3 computing truth tables

SYMBOLIC LOGIC

Computing Truth Values

SYMBOLIC LOGIC

Definition

An assertion is a statement. A proposition is a statement which is either true or false. If a proposition is true we assign the truth value “TRUE” to it. If a proposition is false, we assign the truth value “FALSE” to it. We will denote by “T” or “1”, for the truth value TRUE and by “F" or “0” for the truth value FALSE.

SYMBOLIC LOGIC

Examples

The following are examples of propositions:1. 2 > 42. The billionth prime, when written in base 10,

ends in a 3.3. All men are mortals.4. 9 is a prime number.

SYMBOLIC LOGIC

Non-Examples

The following are non-examples of propositions:1. x < y2. Factor 𝑥2 + 2𝑥 + 1.3. 𝑥 = 104. How old are you?

SYMBOLIC LOGIC

*Decide whether the following are propositions or not:

1. 23 = z2. 10 – 7 = 33. 5 < 274. All women are mammals.5. Where do you live?

SYMBOLIC LOGIC

Definition

A propositional variable, denoted by 𝑃, 𝑄, 𝑅 …denotes an arbitrary proposition with an unspecified truth value.

A propositional variable is a variable that represents a proposition.

SYMBOLIC LOGIC

Do Worksheet 1

SYMBOLIC LOGIC

Definition

Given two propositional variables 𝑃 and 𝑄. These two propositional variables maybe combined to form a new one. These are combined usingthe logical operators or logical connectives : “and”, “or” or “not”.

SYMBOLIC LOGIC

These new proposition are:

1. (Conjunction of P and Q) P and Q, denoted by 𝑃 𝑄;2. (Disjunction of P and Q) P or Q, denoted by 𝑃 𝑄;3. (Negation of P) not P, denoted by ¬ 𝑃.

Truth Tables

𝑃 ¬ 𝑃

1 0

0 1

𝑃 𝑄 𝑃 𝑄 𝑃 𝑄

1 1 1 1

1 0 0 1

0 1 0 1

0 0 0 0

Note: Other books represent the negation of P as ~P.

SYMBOLIC LOGIC

Inclusive and exclusive disjunction:

1. Inclusive disjunction denoted by 𝑃 𝑄 is True when either or both of the disjuncts are True.

2. Exclusive disjuction denoted by P ⋁ Q is True when only one of the disjuncts is true and the other is false.

Truth Table:𝑃 𝑄 𝑃 𝑄 P ⋁ Q

1 1 1 0

1 0 1 1

0 1 1 1

0 0 0 0

SYMBOLIC LOGIC

Do Worksheet 2 & 3

SYMBOLIC LOGIC

Definition

The proposition “P implies Q”, denoted by 𝑃⟹Q is called an implication.

The operand P is called the hypothesis, premise or antecedent while the operand Q is called the conclusion or the consequence

𝑃 𝑄 𝑃⟹Q

0 0 1

0 1 1

1 0 0

1 1 1

SYMBOLIC LOGIC

Definition

Given the implication 𝑃⟹Q , its converse is 𝑄⟹P , its inverseis¬𝑃⟹¬Q, and its contrapositive is ¬𝑄⟹¬P .

The operand P is called the hypothesis, premise or antecedent and the operand Q is called the conclusion or the consequence

𝑃 𝑄 𝑃⟹Q ¬𝑃⟹¬Q 𝑄⟹P ¬𝑄⟹¬P

0 0 1 0 1 1

0 1 1 0 0 1

1 0 0 1 1 0

1 1 1 0 1 1

SYMBOLIC LOGIC

Definition

A biconditional proposition is expressed linguistically by preceding either component by ‘if and only if ’.

The truth table for a biconditional propositional form, symbolised by P ⟷ Q is shown below.

𝑃 𝑄 P ⟷ Q

0 0 1

0 1 0

1 0 0

1 1 1

SYMBOLIC LOGIC

Do Worksheet 4

SYMBOLIC LOGIC

Definition

A propositional form is an assertion which contains at least one propositional variable and maybe generated by the following rules:1. A propositional variable standing alone is a propositional

form;2. If P is a propositional form, then Q is also a propositional

form;3. If P and Q are propositional forms, then 𝑃 𝑄, 𝑃 𝑄, 𝑃 ⟺ 𝑄

are propositional forms;4. A string of symbols containing propositional variables,

connectives and parentheses is a propositional form if and only if it can be obtained by infinitely many applications of rules (1.); (2.) or (3.) above.

Definition

Let X be a set of propositions.A truth assignment (to X) is a function : X {true, false} that assigns to each propositional variable a truth value. (A truth assignment corresponds to one row of the truth table.If a truth value of a compound proposition under truth assignment is true, we say that satisfies P, otherwise we say that falsifies P.A tautology is a propositional form where every truth assignment satisfies P, i.e. All entries of its truth table are true. A contradiction or absurdity is a propositional form where every truth assignment is false;A contingency is a propositional form that is neither tautology nor contradiction.

Examples:P V P is a tautology.P P is a contradiction.

For each of the following compound propositions determine if it is a tautology, contradiction or contingency:1. (p v q) p q2. P v q v r v (p q r)3. (p q) (p v q)

SYMBOLIC LOGIC

Do Worksheet 5

SYMBOLIC LOGIC

Definition

A logically equivalent propositional form have identical values for each assignment of the truth values to their component propositional variables.

We can denote the logical equivalent variables P and Q as:

𝑃 ⟺ 𝑄or

𝑃 ≡ 𝑄

(𝑥 + 2)2 and 𝑥2 + 4𝑥 + 4 are regarded as equivalent algebraic expressions.

Example

Show that P ⟹ Q and ¬P ⋁ Q are logically equivalent propositional forms.

P ⟹ Q and ¬P ⋁ Q are logically equivalent propositional forms.

SYMBOLIC LOGIC

𝑃 𝑄 P ⟹ Q ¬P ¬P ⋁ Q

0 0 1 1 1

0 1 1 1 1

1 0 0 0 0

1 1 1 0 1

Example

Given the propositional forms Q ⋁ ¬P, ¬Q ⟹ ¬P and ¬P ⋀ ¬Q, between which pairs of these forms does the relation logical equivalence exist?

¬Q ⟹ ¬P and ¬P ⋀ ¬Q are logically equivalent

SYMBOLIC LOGIC

𝑃 𝑄 ¬P ¬Q Q ⋁ ¬P ¬Q ⟹ ¬P ¬P ⋀ ¬Q,

0 0 1 1 1 1 0

0 1 1 0 1 1 0

1 0 0 1 0 0 0

1 1 0 0 1 1 1

SYMBOLIC LOGIC

The following are logical identities or rules of replacement.

SYMBOLIC LOGIC

SYMBOLIC LOGIC

SYMBOLIC LOGIC

SYMBOLIC LOGIC

Example

Show that ¬(Q ⋀ P) ⟺ P ⟹¬Q.

Solution

¬(Q ⋀ P) ⟺ ¬Q ⋁¬P (De Morgan’s)⟺ ¬P ⋁¬Q (Commutativity)⟺ P ⟹¬Q (MI)

SYMBOLIC LOGIC

Example

Show that P ⋀ [(P ⋀ Q) ⋁ R] ⟺ P ⋀ (Q ⋁ R).

Solution

P ⋀ [(P ⋀ Q) ⋁ R] ⟺ [P ⋀ (P ⋀ Q)] ⋁ (P ⋀ R) (Dist)⟺ [(P ⋀ P) ⋀ Q] ⋁ (P ⋀ R) (Assoc)⟺ (P ⋀ Q) ⋁ (P ⋀ R) (Indempotence)⟺ P ⋀ (Q ⋁ R) (Dist)

SYMBOLIC LOGIC

Do Worksheet 6

SYMBOLIC LOGIC

Definition

An argument is a collection of propositions wherein it is claimed that one of the propositions, called the conclusion, follows from the other propositions, called the premise of the argument. the conclusion is usually preceded by such words as therefore, hence, then, consequently.

Classification of Arguments:1. Inductive argument is an argument where it is claimed

that within a certain probability of error, the conclusion follows from a premise; and

2. Deductive argument is an argument where is it claimed that the conclusion absolutely follows from the premise.

SYMBOLIC LOGIC

A deductive argument is said to be valid if whenever the premises are all true, then the conclusion is also true. In other words if 𝑃1, 𝑃2, … 𝑃𝑛 are premises and Qis the conclusion of the argument 𝑃1 𝑎𝑛𝑑 𝑃2, 𝑎𝑛𝑑 …𝑃𝑛 therefore Q is valid if and only if the corresponding prepositional form

(𝑃1 𝑃2 … 𝑃𝑛) ⟹ 𝑄,

is a tautology. Otherwise, the argument is invalid.

SYMBOLIC LOGIC

To show that an argument is invalid, we have to show an instance where the conclusion is false and the premises are all true.

Show that the following argument is invalid using Truth Table.

SYMBOLIC LOGIC

To show the validity of arguments, we may use the truth table. However, this method is impractical specially if the argument contains several propositional variables. A more convenient method is by deducing the conclusion from the premises by a sequence of shorter, more elementary arguments known to be valid.

SYMBOLIC LOGIC

Rules of InferenceThese are known valid argument forms.

SYMBOLIC LOGIC

SYMBOLIC LOGIC

SYMBOLIC LOGIC

SYMBOLIC LOGIC

Construct a formal proof of validity of the following arguments:

a) Jack is in Paris only if Mary is in New York. Jack is in Paris and Fred is in Rome. Therefore, Mary is in New York.

b) If Mark is correct then unemployment will rise and if Ann is correct then there will be a hard winter. Ann is correct. Therefore unemployment will rise or there will be a hard winter or both.

SYMBOLIC LOGIC

Solution for (a):

J: Jack is in Paris.M: Mary is in New York.F: Fred Is in Rome.

The premises of the argument are J ⟹ M and J ⋀ F. The conclusion is M.

1. J ⟹ M (premise)2. J ⋀ F (premise)3. J (2. Simp)4. M (1, 3. MP)

SYMBOLIC LOGIC

Solution for (b):

M: Mark is correct.U: Unemployment will rise.A: Ann is correct.H: There will be a hard winter.

The premises of the argument are: (M ⟹ U) ⋀ (A ⟹ H) and A. The conclusion is: U ⋁ H.

1. (M ⟹ U) ⋀ (A ⟹ H) (premise)2. A (premise)3. (A ⟹ H) ⋀ (M ⟹ U) (1. Comm)4. A ⟹ H (3. Simp)5. H (4, 2. MP)6. H ⋁ U (5. Add)7. U ⋁ H (6. Comm)

SYMBOLIC LOGIC

Alternative Solution for (b):

M: Mark is correct.U: Unemployment will rise.A: Ann is correct.H: There will be a hard winter.

The premises of the argument are: (M ⟹ U) ⋀ (A ⟹ H) and A. The conclusion is: U ⋁ H.

1. (M ⟹ U) ⋀ (A ⟹ H) (premise)2. A (premise)3. A ⋁ M (2. Add)4. M ⋁ A (3. Comm)5. U ⋁ H (1, 4. CD)

SYMBOLIC LOGIC

Do Worksheet 7

Definition

A conditional proof is a method of formal proof which is particularly useful in establishing the validity of an argument. The argument has a conclusion which can be expressed as a conditional proposition.

Consider an argument form with premises p1, p2,…,pn and conclusion q r. Note that this argument form is valid if and only if (p1 p2…pn ) (q r).

Now the exportation replacement rule states that p (q r) (p q) r.So that the validity of the condition

(p1 p2…pn ) (q r) is a tautology can be replaced by (p1 p2…pnq) r is a tautology.

SYMBOLIC LOGIC

Example

Prove the validity of the following arguments using the method of conditional proof.

1. If we have a party then we’ll invite Lana and Bob. If we invite Lana or Bob then we must invite Jake. Therefore if we have a party then we must invite Jake.

Solution

We symbolize the following simple propositions:

P: We have a party B: We’ll invite Bob.L: We’ll invite Lana. J: We must invite Jake.

SYMBOLIC LOGIC

(Continued)

The premises of the argument are:P ⟹ (L ⋀ B) and (L ⋁ B) ⟹ J

The conclusion is the conditional P ⟹ J.

Proof:1. P ⟹ (L ⋀ B) (premise)2. (L ⋁ B) ⟹ J (premise)3. P (CP)4. L ⋀ B (1, 3. MP)5. L (4. Simp)6. L ⋁ B (5. Add)7. J (2, 6. MP)8. P ⟹ J (3 – 7. CP)

SYMBOLIC LOGIC

Example

2. If we invite Lana then Jake will sulk, and if we invite Bob then Alice will leave. So if we invite Lana and Bob then Jake will sulk and Alice will leave.

Solution

We symbolize the following simple propositions:

L: We invite Lana. B: We invite Bob.J: Jake will sulk. A: Alice will leave.

The premise of the argument is: (L ⟹ J) ⋀ (B ⟹ A)and the conclusion is (L ⋀ B) ⟹ (J ⋀ A).

SYMBOLIC LOGIC

Solution

Proof:1. (L ⟹ J) ⋀ (B ⟹ A) (premise)2. L ⋀ B (CP)3. L ⟹ J (1. Simp)4. (B ⟹ A) ⋀ (L ⟹ J) (1. Com)5. B ⟹ A (4. Simp)6. L (2. Simp)7. J (3, 6. MP)8. B ⋀ L (2. Com)9. B (8. Simp)10. A (5, 9. MP)11. J ⋀ A (7, 10. Conj)12. (L ⋀ B) ⟹ (J ⋀ A) (2 – 11. CP)

SYMBOLIC LOGIC

Do Worksheet 8