chapter 1 introduction to logic
DESCRIPTION
A beamer presentation I used in my Math 29 Basic Concepts in Mathematics course. The presentation was based on certain books i.e. Rosen, Barnier and Gerstein.TRANSCRIPT
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Math 29: Basic Concepts in Mathematics
Reymart S. Lagunero
Departamento ng Matematika at Agham Pangkompyuter
Unibersidad ng Pilipinas Baguio
04 Agosto 2015
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Chapter 1: Introduction to Logic
What is Logic?
What is Logic?
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Chapter 1: Introduction to Logic
What is Logic?
What is Logic?
Definition
Logic is the study of the methods and principles used to distinguish good(correct) from bad (incorrect) reasoning. There are objective criteria withwhich correct reasoning may be defined. If these criteria are not known,then they cannot be used. The aim of the study of logic is to discover andmake available those criteria that can be used to test arguments, and to
sort good arguments from bad ones.
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Chapter 1: Introduction to Logic Statements
Statements
Definition
We define a statement intuitively as a sentence that can be assigned
either to the class of things we would call TRUE or to the class of thingswe would call FALSE, but not both. The TRUTH VALUE of a statementis “true” if the statement is true and “false” if the statement is false.
We often use P or Q to denote statements, or perhaps P 1, P 2, . . . , P n if
there are several statements involved.
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Chapter 1: Introduction to Logic Statements
Examples
Example
Some examples of statements are:
1 Today is Thursday.
2 I passed the final exam in Mathematics 53.
3 2 is a prime number.
4 There is an integer between 1 and 2.
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Chapter 1: Introduction to Logic Statements
Examples
Example
The sentence “The real number r is rational.” is a statement provided weknow precisely what real number r is being referred to. Without thisadditional information, however, it is impossible to judge whether thissentenced is true or false. Such a sentence is often referred to as an opensentence or an indeterminate.
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Ch 1 I d i L i S
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Chapter 1: Introduction to Logic Statements
Examples
Example
We do not want to consider paradoxes as statements. For example “Thissentence is false.” cannot be either true or false. If you think this sentenceis true, then it is false. But, if it is false, then it is true. This sentence iscalled the liar paradox.
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Ch t 1 I t d ti t L i L i l C ti
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Chapter 1: Introduction to Logic Logical Connectives
Logical Connectives
Complex statements can be constructed from simple ones by means of logical connectives. Five logical connectives will be considered in thissection, namely: ¬, ∨, ∧, =⇒ and ⇐⇒ .
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Chapter 1: Introduction to Logic Logical Connectives
Negation
Definition
Negation The negation of a statement P is the statement ¬P , read as
“not P ”. The negation ¬P is true when P is false, and false when P istrue, that is
P ¬P
T F F T
(1.1)
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Chapter 1: Introduction to Logic Logical Connectives
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Chapter 1: Introduction to Logic Logical Connectives
Example
Example
P ¬P
It is raining today. It is not raining today.
2 is a prime number. 2 is not a prime number.2 is a composite number.
My shirt is black. My shirt is not black.
Observe in the last example that negating the statement “My shirt is
black” into “My shirt is white” is not logically correct because not wearinga black shirt does not necessarily mean that a person is wearing a whiteshirt (perhaps, a pink shirt!).
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Chapter 1: Introduction to Logic Logical Connectives
Conjunction
Definition
Conjunction The conjunction of two statements P and Q is thestatement P ∧ Q, read as “P and Q”. The conjuction P ∧ Q is true only
if both P and Q are true; otherwise, P ∧ Q is false, that is,
P Q P ∧ Q
T T T T F F
F T F F F F
(1.2)
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Chapter 1: Introduction to Logic Logical Connectives
Example
Example
Suppose that
P : John is a sophomore.
andQ : 2 is less than 1.
Then P ∧ Q is the statement “John is a sophomore and 2 is less than 1.”
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p g g
Disjunction
Definition
Disjunction The disjunction of two statements P and Q is the statementP ∨ Q, read as “P or Q”. The disjunction P ∨ Q is true if at least one of
P and Q is true; otherwise, P ∨ Q is false, that is,
P Q P ∨ Q
T T T T F T
F T T F F F
(1.3)
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g g
Example
Example
From the last example, P ∨ Q is the statement “John is a sophomore or 2is less than 1.”
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Exclusive Disjunction
Remark
To address a statement involving P and Q that is true when precisely oneof them is true, we use the exclusive disjunction, denoted by P ∨̇Q, that
is,P Q P ∨̇ Q
T T F T F T F T T F F F
(1.4)
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Implication
Definition
Implication For statements P and Q, the implication is the statement: “If P , then Q”, and is denoted by P =⇒ Q. The truth table for P ⇒ Q is
given byP Q P ⇒ Q
T T T T F F F T T F F T
(1.5)
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Example
Example
A student is taking the Math 29 class and is currently receiving a B+. Heconsults his instructor a few days before the final exam and asks him, “Is
there any chance that I can get an A in this course?” His instructor looksthrough his grade book and says, “If you earn an A in the final exam, thenyou will receive a A for your final grade.”
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Example
Example
A student is taking the Math 29 class and is currently receiving a B+. Heconsults his instructor a few days before the final exam and asks him, “Is
there any chance that I can get an A in this course?” His instructor looksthrough his grade book and says, “If you earn an A in the final exam, thenyou will receive a A for your final grade.” We now check the truth orfalseness of this implication based on the various combinations of truthtables of the statements P : You earn an A on the final exam and Q: Youreceive a A for your final grade, which make up the implication.
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...continuation: Example
Suppose first that P and Q are both true, that is, the student receives an
A on his final exam and later learns that he got an A for his final grade inthe course. Did his instructor tell the truth?
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...continuation: Example
Suppose first that P and Q are both true, that is, the student receives an
A on his final exam and later learns that he got an A for his final grade inthe course. Did his instructor tell the truth? I think we all agree thatshe did! So if P and Q are true, then so is P ⇒ Q, which agrees with thefirst row of the truth table.
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...continuation: Example
Suppose that P is true and Q is false. So the student got a A on his final
exam but did not receive a A as final grade, say he received B. Certainlyhis instructor did not do as she promised. What he said was false, whichagrees with the second row of the truth table.
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...continuation: Example
Suppose that P is false and Q is true. In this case, the student did not getan A on his final exam, say he earned B, but when he received his final
grades, he learned that his final grade was an A. How could thishappen?
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...continuation: Example
Suppose that P is false and Q is true. In this case, the student did not getan A on his final exam, say he earned B, but when he received his final
grades, he learned that his final grade was an A. How could thishappen? Perhaps his instructor made a mistake, or perhaps the finalexam was unusually difficult, and a grade of B on it was exceptionallygood performance. In either case, the instructor did not lie; he told thetruth. This agrees with the third row of the table.
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...continuation: Example
Suppose that P and Q are both false. That is, suppose the student didnot get an A on his final exam, and he also did not get an A for a finalgrade. The instructor did not lie here either. He promised nothing if thestudent did not get an A on the final exam. So the instructor told thetruth, and this agrees with the fourth row of the table.
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Indeterminates
We have concerned ourselves only with statements. In mathematics,however, we are often interested in sentences containing variables andwhose truthfulness or falseness is only known once we have assigned values
to the variables, previously known as indeterminate. Just as newstatements can be formed from two statements P and Q by taking theirnegation, conjunction, or disjunction, new indeterminate can be formedfrom two open sentences P and Q by taking their negation, conjunction,or disjunction. Surprisingly, P ⇒ Q could be a statement even though P
and Q are not statements.
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Example
Example
Consider the indeterminate p : x = −3 and q : |x|= 3, where x ∈ R.Indeed, it would be more appropriate to write
p(x) : x = −3 and q (x) : |x|= 3.
In this case, we have p(x) ⇒ q (x) can be stated as
If x = −3, then |x|= 3.
This implication is, in fact, a true statement.
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Biconditional
Definition
Biconditional The biconditional of P and Q, denoted by P ⇐⇒ Q, istrue precisely when both P and Q have the same truth values, that is,
P Q P ⇔ Q
T T T T F F F T F
F F T
(1.6)
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Example
Suppose thatP : An integer is divisible by 6.
andQ : An integer is divisible by 2 and 3.
Then P ⇔ Q is the statement “An integer is divisible by 6 if and only if itis divisible by 2 and 3.”
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Rules of Precedence
For statements expressed symbolically, punctuation is accomplished byparentheses. To keep statements from looking cluttered, we will usecertain conventions for leaving out parentheses.
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Rules of Precedence
For statements expressed symbolically, punctuation is accomplished byparentheses. To keep statements from looking cluttered, we will usecertain conventions for leaving out parentheses. The hierarchial order of the logical connectives is as follows: ¬, then ∨ or ∧ have the sameprecedence, followed by =⇒ , and then ⇐⇒ .
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Rules of Precedence
The following is a list of ambiguous statements and thus, groupingsymbols have to be present when writing them.
1 ¬P ∧ Q does not stand for ¬(P ∧ Q); it stands for (¬P ) ∧ Q.
2 P ∨ Q ⇒ Q stands for (P ∨ Q) ⇒ R, not for P ∨ (Q ⇒ R).3 P ∨ R ∧ T is ambiguous, since it is not clear whether to apply ∨ or ∧
first.
4 P ⇒ Q ⇒ R is ambiguous, since either occurrence of ⇒ can be
applied first.
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Tautologies, Contradictions and Logical Equivalence
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Prime, Composite, and Component Statements
Statements without logical connectives are called prime statements while
statements with connectives are called composite statements. Thestatements which formed a composite statement are called its componentstatements.
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Tautologies, Contradictions and Logical Equivalence Tautology
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Definition (Tautology)
A statement S is called a tautology if it is true for all possiblecombinations of truth values of the component statements that composeS .
Remark
The symbol I will denote a statement that always has truth value T .
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Tautologies, Contradictions and Logical Equivalence Tautology
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Example
Prove that the following compound statements is a tautology.
1 P ∨ (¬P ).
2 (¬Q) ∨ (P ⇒ Q).
3 P ∧ (P ⇒ Q) ⇒ Q (Modus Ponens)
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Tautologies, Contradictions and Logical Equivalence Tautology
E i 5 Mi
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Exercise: 5 Minutes
1 Construct the truth table of the following complex statement:
¬(P ∨ Q) ∧ (P ∧ (¬Q ∨ (¬P ∨ Q))) .
When is the whole statement true?
2 When will the whole statement be true given
¬(P ∧ Q) ∨ R =⇒ ¬P ∨ R.
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Tautologies, Contradictions and Logical Equivalence Contradictions
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Definition (Contradiction)A statement S is called a contradiction if it is false for all possiblecombinations of truth values of the component statements that are usedto form S .
Remark
In fact, if a compound statement S is a tautology, then the compoundstatement ¬S is a contradiction. The symbol O will denote a statementthat always has truth value F .
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Tautologies, Contradictions and Logical Equivalence Contradictions
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Example
Prove that the following compound statements is a contradiction.
1
P ∧ (¬P ).2 (P ∧ Q) ∧ (Q ⇒ (¬P )).
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Tautologies, Contradictions and Logical Equivalence Logical Equivalence
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Definition (Logical equivalence)
We say that statements P and Q are logically equivalent, denoted by
P ≡ Q, if they have the same truth values for all combinations of truthvalues of their component statements.
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Tautologies, Contradictions and Logical Equivalence Logical Equivalence
Theorem
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Theorem
It is clear that logical equivalence is reflexive, commutative and transitive.
We list some of the well-known logical equivalence in the followingtheorem.
Theorem
For statements P , Q and R, we have
Commutative Laws
P ∨ Q ≡ Q ∨ P P ∧ Q ≡ Q ∧ P
Associative Laws
P ∨ (Q ∨ R) ≡ (P ∨ Q) ∨ R P ∧ (Q ∧ R) ≡ (P ∧ Q) ∧ R)
Distributive Laws
P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R) P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
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Tautologies, Contradictions and Logical Equivalence Logical Equivalence
continuation: Theorem
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...continuation: Theorem
DeMorgan’s Laws
¬(P ∨ Q) ≡ (¬P ) ∧ (¬Q) ¬(P ∧ Q) ≡ (¬P ) ∨ (¬Q)
Rule of Double Negation
P ≡ ¬(¬P )
Or-form of an ImplicationP ⇒ Q ≡ (¬P ) ∨ Q
Contrapositive of an Implication
P ⇒ Q ≡ (¬Q) ⇒ (¬P )
Rule for Direct Proof
(P ∧ R) ⇒ Q ≡ P ⇒ (R ⇒ Q)
Biconditional
P ⇔ Q ≡ (P ⇒ Q) ∧ (Q ⇒ P )Reymart S. Lagunero (UP na, Baguio pa!) Math 29: Introduction to Logic 04 Agosto 2015 35 / 91
Tautologies, Contradictions and Logical Equivalence Logical Equivalence
continuation: Theorem
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...continuation: Theorem
Adjunction
(P ⇒ R) ∧ (P ⇒ Q) ≡ (P ⇒ (R ∧ Q))
Rule for proof by contradiction
(P ∧ ¬Q) ⇒ O ≡ (P ⇒ Q)
The symbol ≡ has the least precedence among the logical connectivesintroduced above.
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Tautologies, Contradictions and Logical Equivalence Quantified Statements
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Definition (Universal quantifier)
The universal quantifier ∀x means “for all x” or “for any x.”
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Tautologies, Contradictions and Logical Equivalence Quantified Statements
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Example
If x is a real number, then x2 ≥ 0. This statement is an implication, of course, and can be rephrased by
The square of every real number is nonnegative
orFor every real number x, we have x2 ≥ 0.
If we define the open sentence P (x) by P (x) : x2 ≥ 0, then we can rewritethe above statement in terms of the universal quantifier as
(∀x ∈ R)(P (x)).
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Notice that if the set of complex numbers were under consideration, thenthe statement in the previous example would be false, since i2 = −1. Sothe truth value of a quantified statement may depend on the priorspecification of a set from which all elements are understood to come.
Such a set is called the universal set. In general, the statement(∀x ∈ U )(P (x)) is true if P (a) is true for all substitutions a ∈ U . If auniversal set U is not explicitly stated, the student can assume thatU = R.
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Definition (Existential quantifier)The existential quantifier ∃x means “there exists an x such that” or“there is an x such that.”
Example
There exists a real number x such that x2 = 3. If we let P (x) : x2 = 3then this statement can be rewritten in terms of the existential quantifieras
(∃x ∈ R)(P (x)).
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In general, the statement (∃x ∈ U )(P (x)) is true if there is at least onesubstitution a ∈ U for which P (a) is true.Some statements contain more than one quantifiers. As you will see, theorder of quantifiers is important. For instance, the statement
(∃u)(∀x)(P (x, u)) is read as “There exists an element u such that for allx, we have P (x, u)” while the statement (∀x)(∃u)(P (x, u)) represent thestatement “For every x there exists an element u such that P (x, u).”
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Example
The statement “Every person has a mother” is of the form
(∀x)(∃y)(y is the mother of x)
while the statement “Someone is the mother of everyone” is of the form(∃y)(∀x)(y is the mother of x).
If the universal set is the set of all people who have ever been alive, thefirst statement is true, but the second is false.
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Assignment
For each of the following statements, use your creativity and constructseveral statements that are logically equivalent to the given statement.
1 No elements of the set A exceeds m.
2 Some element of the set A exceeds m.
3 A contains an element that is greater than every element of B.
4 Every element of A is greater than every element of B.
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Tautologies, Contradictions and Logical Equivalence Negating Statements
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Using DeMorgan’s laws, we saw that
¬(P ∨ Q) ≡ (¬P ) ∧ (¬Q) and ¬(P ∧ Q) ≡ (¬P ) ∨ (¬Q).
Using the or-form of an implication, we see that
¬(P ⇒ Q) ≡ ¬(¬P ∨ Q) ≡ P ∧ (¬Q),
while using the fact that P ⇔ Q ≡ (P ⇒ Q) ∧ (Q ⇒ P ), we have
¬(P ⇔ Q) ≡ ¬[(P ⇒ Q) ∧ (Q → P )] ≡ (P ∧ (¬Q)) ∨ (Q ∧ (¬P )).
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Tautologies, Contradictions and Logical Equivalence Negating Statements
Now, we consider negating statements with quantifiers. Suppose someonek th f ll i l i
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makes the following claim
Every student in the class passed the first exam.
For you to deny this claim, what sort of statement would you make? Youwould probably say something like
At least one person in the class failed the first exam.
If we let U be the set of students in the class and let Q be the set of students who passed the first exam. We can write the original statementin terms of the universal quantifier as
(∀x ∈ U )(x ∈ Q)
and its negation would be
(∃x ∈ U )(x /∈ Q)
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Tautologies, Contradictions and Logical Equivalence Negating Statements
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In general, we can say that
¬[(∀x ∈ U )(P (x))] ≡ (∃x)(¬P (x)).
So the trick to negating a universal statement is that the ¬ symbol crawlsover the ∀x ∈ U and converts it to ∃x as it goes.
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Tautologies, Contradictions and Logical Equivalence Negating Statements
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Example
Write in terms of the quantifiers the negation of the following statements.
1 For all x ∈ Z, x2 ≥ 0.
2 For all x ∈ Z, if x is divisible by 6 then x is divisible by 3.3 For all x ∈ R and x > 1, we have x3 − x > 0.
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Tautologies, Contradictions and Logical Equivalence Negating Statements
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Now we negate statements involving the existential quantifier. Considerthe following statement
There is a passenger traveling in the jeepney who didn’t pay his fare.
If we let U be the set of all passengers riding in the jeepney and R be the
set of passengers who didn’t pay his fare, in terms of the existentialquantifier, the statement becomes
(∃x ∈ U )(x ∈ R).
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Tautologies, Contradictions and Logical Equivalence Negating Statements
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To negate this, we could say
Every passenger in the jeepney paid his fare
which is equivalent to(∀x ∈ U )(x /∈ R).
In general, we can say that
(∃x ∈ U )(P (x)) ≡ (∀x ∈ U )(¬P (x)).
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Tautologies, Contradictions and Logical Equivalence Negating Statements
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Example
Write in terms of the quantifiers the negation of the following statements.
1 Someone in this class cheated on the final exam.
2 There exists a natural number x such that x ≤ y for all y ∈ N .
3 For every > 0 there exists a δ > 0 such that,
if 0 < |x − a|< δ , then |f (x) − L|< .
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Tautologies, Contradictions and Logical Equivalence Negating Statements
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Sometimes it is important to know not only that something exists, butalso that exactly one such thing exists. If there exists exactly one thingwith a certain property, we say that it exists uniquely. The mathematicalstatement
(∃! x ∈ U )(P (x))is read “There exists a unique x such that P (x).” The standard way of defining existence is the following.
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Tautologies, Contradictions and Logical Equivalence Negating Statements
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Definition (Unique existence)We say that there exists a unique x with property P provided that
1 there exists x ∈ U with property P , and
2 for all x1 ∈ U and x2 ∈ U , if x1 and x2 both have property P , then
x1 = x2.In other words
[(∃x ∈ U )(P (x))] ∧ [(∀x1 ∈ U )(∀x2 ∈ U )(P (x1) ∧ P (x2)) ⇒ (x1 = x2)].
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Tautologies, Contradictions and Logical Equivalence Disproving a Statement
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Sometimes we are presented with a mathematical statement and we areasked to verify whether it is a tautology. Of course, you can create a truthtable for the given statement and check if it is indeed a tautology, that is,if all entries in the main column are T . The combination of truth valuesassigned to the component statements in any row that produces a false inthe main column of the truth table is called a counterexample.
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Example
Show that (P ⇒ Q) ∧ Q ⇒ P is not a tautology.
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Example
Show that (P ⇒ Q) ∧ Q ⇒ P is not a tautology.
P Q (P ⇒ Q) ∧ Q ⇒ P
T T T T F T
F T F F F T
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Tautologies, Contradictions and Logical Equivalence Disproving a Statement
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Example
Show that (P ⇒ Q) ∧ Q ⇒ P is not a tautology.
P Q (P ⇒ Q) ∧ Q ⇒ P
T T T T F T
F T F F F T
The counterexample is P false and Q is true. Claiming the above exampleto be a tautology is a well-known fallacy of logic. It is called the fallacy of
asserting the conclusion, since the conclusion Q has been asserted aspart of the hypothesis.
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Tautologies, Contradictions and Logical Equivalence Disproving a Statement
Counterexamples will become important to us in disproving a givenmathematical statement. To disprove a given mathematical statement, we
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mathematical statement. To disprove a given mathematical statement, weprove that its negation is a tautology. This is where our techniques of negation come in handy.
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Tautologies, Contradictions and Logical Equivalence Disproving a Statement
Counterexamples will become important to us in disproving a givenmathematical statement. To disprove a given mathematical statement, we
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p g ,prove that its negation is a tautology. This is where our techniques of negation come in handy.
Example
Consider the statement
For all sets A, B, and C , if A ∪ C = B ∪ C , then A = B.
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Tautologies, Contradictions and Logical Equivalence Disproving a Statement
Counterexamples will become important to us in disproving a givenmathematical statement. To disprove a given mathematical statement, we
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p g ,prove that its negation is a tautology. This is where our techniques of negation come in handy.
Example
Consider the statement
For all sets A, B, and C , if A ∪ C = B ∪ C , then A = B.
We can disprove the statement by proving its negation to be true, that is,we would want to show that the statement
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Tautologies, Contradictions and Logical Equivalence Disproving a Statement
Counterexamples will become important to us in disproving a givenmathematical statement. To disprove a given mathematical statement, we
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p gprove that its negation is a tautology. This is where our techniques of negation come in handy.
Example
Consider the statement
For all sets A, B, and C , if A ∪ C = B ∪ C , then A = B.
We can disprove the statement by proving its negation to be true, that is,we would want to show that the statement
There exists sets A, B, and C such that A ∪ C = B ∪ C and A = B
is true.
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Tautologies, Contradictions and Logical Equivalence Disproving a Statement
Counterexamples will become important to us in disproving a givenmathematical statement. To disprove a given mathematical statement, we
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prove that its negation is a tautology. This is where our techniques of negation come in handy.
Example
Consider the statement
For all sets A, B, and C , if A ∪ C = B ∪ C , then A = B.
We can disprove the statement by proving its negation to be true, that is,we would want to show that the statement
There exists sets A, B, and C such that A ∪ C = B ∪ C and A = B
is true. For instance, let A = {1}, B = {2}, and C = {1, 2}. ThenA ∪ C = B ∪ C , but A = B . Again, such an example is also called acounterexample.
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Tautologies, Contradictions and Logical Equivalence Translating English Sentences to Logical Expressions
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Translating English Sentences to Logical Expressions
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Tautologies, Contradictions and Logical Equivalence Translating English Sentences to Logical Expressions
Example
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Translate the following English sentences into logical expressions.1 You can only access the Internet from campus only if you are a
computer science major or you are not a freshman.
Solution:
We let a, c, and f represent “You can access the Internet from campus”,“You are a computer science major”, and “You are a freshman”,
respectively. Noting that “only if” is one way a conditional statement canbe expressed, this sentence can be represented as a =⇒ (c ∨ ¬f ).
2 You cannot ride the roller coaster if you are under 4 feet tall unlessyou are older than 16 years old.
Solution:
Let q , r, and s represent “You can ride the roller coaster”, “You are under4 feet tall” and “You are older than 16 years old”, respectively. Then thesentence can be translated to (r ∧ ¬s) =⇒ ¬q .
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Tautologies, Contradictions and Logical Equivalence Logic Puzzles
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Logic Puzzles
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Tautologies, Contradictions and Logical Equivalence Logic Puzzles
Example
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Smullyan posed many puzzles about an island that has two kinds of inhabitants, knights, who always tell the truth, and their opposites, knaves,who always lie. You encounter two people A and B. What are A and B if A says “B is a knight” and B says “The two of us are opposite types?”
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Tautologies, Contradictions and Logical Equivalence Logic Puzzles
Example
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A father tells his two children, a boy and a girl, to play in their backyardwithout getting dirty. However, while playing, both children get mud ontheir foreheads. When the children stop playing, the father says “At leastone of you has a muddy forehead,” and then asks the children to answer
“Yes” or “No” to the question: “Do you know whether you have a muddyforehead?” The father asks this question twice. What will the childrenanswer each time this question is asked, assuming that a child can seewhether his or her sibling has a muddy forehead, but cannot see his or herown forehead? Assume that both children are honest and that the children
answer each question simultaneously.
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Tautologies, Contradictions and Logical Equivalence Logic Puzzles
Example
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Four friends have been identified as suspects for an unauthorized accessinto a computer system. They have made statements to the investigatingauthorities. Alice said “Carlos did it.” John said “I did not do it.” Carlossaid “Diana did it.” Diana said “Carlos lied when he said that I did it.”
If the authorities also know that exactly one of the four suspects istelling the truth, who did it? Explain your reasoning.
If the authorities also know that exactly one is lying, who did it?Explain your reasoning.
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Tautologies, Contradictions and Logical Equivalence Logic Puzzles
Example
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Vhong, Jinggy, and Juanito are all enrolled in Math 29. If Vhong comeslate then so is Jinggy. Either Jinggy or Juanito comes late but never both
at the same time. Either Vhong or Juanito or both are always late. If Vhong is on time then Juanito is also on time. Who is always on time?Justify your answer using truth table.
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Rules of Inference for Propositional Logic
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Rules of Inference for Propositional Logic
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Rules of Inference for Propositional Logic
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In this section, we introduce proof techniques and apply them to provingstatements from a collection of premises.
Definition
A proof is a step-by-step demonstration that a statement can be derivedfrom a collection of premises. A premise is a statement that is assumed inthe context of a proof.
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Rules of Inference for Propositional Logic Modus Ponens
Rule of Inference: Modus Ponens
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From P and P implies Q, infer Q. In other words,
P ∧ (P =⇒ Q) =⇒ Q
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Rules of Inference for Propositional Logic Modus Ponens
Rule of Inference: Modus Ponens
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Example
1 Premises: T , T =⇒ S . Prove S .
2 Premises: T =⇒ R, R =⇒ S , T . Prove S .
3 Premises:x is even or x is odd.If x is even or x is odd, then x is not even implies that x is odd.x is not even.
Prove that x is odd.
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Rules of Inference for Propositional Logic Modus Ponens
Rule of Inference: Modus Ponens
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Exercise
Premises:
¬(R ∧ T ) =⇒ ¬R ∨ ¬P
¬(R ∧ T )
¬R
¬R ∨ ¬T =⇒ (¬R =⇒ T )
Prove: T .
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Rules of Inference for Propositional Logic Adjoining Premises
Rule of Inference: Adjoining Premises
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Any statement can be adjoined to the premises in a proof if it can beproved from the premises.
Example
Premises:
P ∨ Q =⇒ (¬P =⇒ Q)P ∨ Q
¬P
Q =⇒ P
Q =⇒ RProve: R.
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Rules of Inference for Propositional Logic Adjoining Premises
Exercise
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Example
Premises:
P ∨ Q =⇒ (¬P =⇒ Q)
P ∨ Q¬P
Q =⇒ R
R =⇒ S ∧ T
Prove: S ∧ T .
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Rules of Inference for Propositional Logic Adjoining Premises
Remark
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We say that P is provable from a set of premises if there is a proof of P
from the set of premises.
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Rules of Inference for Propositional Logic Direct Proof of an Implication
Rule of Inference: Direct Proof of an Implication
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To prove an implication P =⇒ Q from a set of premises R, it is sufficientto assert the hypothesis (Hyp) P as an additional premises and show thatthe conclusion Q is provable from the augmented set of premises. DPI is
based on the tautology
(R =⇒ (P =⇒ Q)) ⇐⇒ (R ∧ P =⇒ Q) .
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Rules of Inference for Propositional Logic Direct Proof of an Implication
Example
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Example
Premises:
x is even or x is odd.
If x is odd, then x2 is not even.
If x is even or x is odd, then x is not even implies that x is odd.
Prove: If x is not even, then x2 is not even.
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Rules of Inference for Propositional Logic Direct Proof of an Implication
Exercise
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Premises:
P
Q =⇒ (P =⇒ ¬R)
Prove: Q =⇒ ¬R.
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Rules of Inference for Propositional Logic Adjunction
Rule of Inference: Adjunction
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If P and Q are provable from the same set of premises, then P ∧ Q is
provable from that set of premises. Adjunction is based on the tautology
(R =⇒ P ) ∧ (R =⇒ Q) ⇐⇒ (R =⇒ P ∨ R)
Example
Premises:
P =⇒ Q
Q ∧ (P ∨ R) =⇒ S ∧ T
P P =⇒ P ∨ R
Prove: S ∧ T .
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Rules of Inference for Propositional Logic Adjunction
Exercise
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Premises:
Q =⇒ P
QP ∧ Q =⇒ R ∨ T
Prove: R ∨ T .
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Rules of Inference for Propositional Logic Substitution
Rule of Inference: Substitution
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Assume that P 2 is obtained from P 1 by substituting R for any occurrence
of S in P 1. Then we can derive P 2 from S ⇐⇒ R and P 1. In otherwords, we may substitute R for S using the premise S ⇐⇒ R. We mayalso substitute S for R whenever S ⇐⇒ R is a premise.
Example
Premises:
R ∨ Q ⇐⇒ P ∨ Q
R ∨ Q =⇒ S
P =⇒ P ∨ Q
P
Prove: S .
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Rules of Inference for Propositional Logic Substitution
Exercise
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Premises:
Q ∨ P ⇐⇒ (¬Q =⇒ P )
¬Q
R =⇒ (Q ∨ P )
R
Prove: P .
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Rules of Inference for Propositional Logic Contradiction
Rule of Inference: Contradiction
To prove Q from a set of premises, it is sufficient to use ¬Q as an
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p Q p , Qadditional premise to prove any contradiction, for example, a statement of the form R ∧ ¬R. Contradiction is based on the tautology
(P ∧ ¬Q) =⇒ O ⇐⇒ (P =⇒ Q)
ExamplePremises:
If Kathy is not the murderer, then Mary is the murderer.
If Kathy is the murderer, then Jane is an accomplice.
If Mary is the murderer, then Noreen does not have an alibi.Noreen has an alibi.
Prove: Kathy is the murderer.
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Rules of Inference for Propositional Logic Contradiction
Exercise
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Premises:
¬(P ∨ Q) =⇒ ¬P ∧ ¬Q
¬P =⇒ R
¬Q =⇒ ¬R
¬P ∧ ¬Q =⇒ ¬P
¬P ∧ ¬Q =⇒ ¬Q
Prove: P ∨ Q.
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Rules of Inference for Propositional Logic Contradiction
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Remark
If we can prove P from a set of premises, then we say that P logicallyfollows, or simply follows, from the set of premises. An argument is asequence of steps proposed as a proof. If an argument is, in fact, a proof,
then it is called a valid argument; if it is not a proof, then it is called aninvalid argument.
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Rules of Inference for Propositional Logic Contradiction
Additional Examples
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State which rule of inference is the basis of the following arguments:
It is below freezing now. Therefore, it is either below freezing orraining now.
It is below freezing and raining now. Therefore, it is below freezingnow.
If it rains today, then we will not have a barbecue today. If we do nothave a barbecue today, then we will have a barbecue tomorrow.Therefore, if it rains today, then we will have a barbecue tomorrow.
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Rules of Inference for Propositional Logic Contradiction
Additional Examples
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Show that the premises “It is not sunny this afternoon and it is colderthan yesterday”, “We will go swimming only if it is sunny”, “If we donot go swimming, then we will take a canoe trip”, and “If we take acanoe trip, then we will be home by sunset” lead to the conclusion
“We will be home by sunset”.Show that the premises “If you send me an e-mail message, then Iwill finish writing the program,” “If you do not send me an e-mailmessage, then I will go to sleep early,” and “If I go to sleep early, thenI will wake up feeling refreshed” lead to the conclusion “If I do not
finish writing the program, then I will wake up feeling refreshed.”
Reymart S Lagunero (UP na Baguio pa!) Math 29: Introduction to Logic 04 Agosto 2015 82 / 91
Rules of Inference for Propositional Logic Contradiction
Additional Examples
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Is the following argument valid?
“If you do every problem in this book, then you will learn discretemathematics.You learned discrete mathematics. Therefore, you did every
problem in this book.”
Is it correct to assume that you did not learn discrete mathematics if youdid not do every problem in the book, assuming that if you do everyproblem in this book, then you will learn discrete mathematics?
Reymart S Lagunero (UP na Baguio pa!) Math 29: Introduction to Logic 04 Agosto 2015 83 / 91
Rules of Inference for Quantified Statements Universal Instantiation
Universal Instantiation
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Universal instantiation is the rule of inference used to conclude thatP (c) is true, where c is a particular member of the domain, given the
premise ∀xP (x). Universal instantiation is used when we conclude fromthe statement “All women are wise” that “Lisa is wise,” where Lisa is amember of the domain of all women.
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Rules of Inference for Quantified Statements Universal Generalization
Universal Generalization
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Universal generalization is the rule of inference that states that ∀xP (x)is true, given the premise that P (c) is true for all elements c in thedomain. Universal generalization is used when we show that ∀xP (x) istrue by taking an arbitrary element c from the domain and showing that
P (c) is true. The element c that we select must be an arbitrary, and not aspecific, element of the domain. That is, when we assert from ∀xP (x) theexistence of an element c in the domain, we have no control over c andcannot make any other assumptions about c other than it comes from thedomain.
Reymart S Lagunero (UP na Baguio pa!) Math 29: Introduction to Logic 04 Agosto 2015 85 / 91
Rules of Inference for Quantified Statements Existential Instantiation
Existential Instantiation
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Existential instantiation is the rule that allows us to conclude that thereis an element c in the domain for which P (c) is true if we know that∃xP (x) is true. We cannot select an arbitrary value of c here, but rather it
must be a c for which P (c) is true. Usually we have no knowledge of whatc is, only that it exists. Because it exists, we may give it a name (c) andcontinue our argument.
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Rules of Inference for Quantified Statements Existential Generalization
Existential Generalization
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Existential generalization is the rule of inference that is used to concludethat ∃xP (x) is true when a particular element c with P (c) true is known.
That is, if we know one element c in the domain for which P (c) is true,then we know that ∃xP (x) is true.
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Combining Rules of Inferences Universal Modus Ponens
Universal Modus Ponens
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This rule tells us that if ∀x(P (x) =⇒ Q(x)) is true, and if P (a) is truefor a particular element a in the domain of the universal quantifier, then
Q(a) must also be true. To see this, note that by universal instantiation,P (a) =⇒ Q(a) is true. Then, by modus ponens, Q(a) must also be true.
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Combining Rules of Inferences Universal Modus Tollens
Universal Modus Tollens
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Universal modus tollens combines universal instantiation and modus
tollens.
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Combining Rules of Inferences Universal Modus Tollens
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Salamat sa pakikinig!
♥
R S L (UP B i !) M h 29 I d i L i 04 A 2015 90 / 91
Combining Rules of Inferences Universal Modus Tollens
Math 29: Basic Concepts in Mathematics
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Reymart S. Lagunero
Departamento ng Matematika at Agham Pangkompyuter
Unibersidad ng Pilipinas Baguio
04 Agosto 2015
R S L (UP B i !) M h 29 I d i L i 0 A 201 91 / 91