equivalence and implication

OutlineProperties of Propositions

Logical EquivalencesPractice with Boolean Operators and Algebra

ImplicationSummary

Equivalence and Implication

Alice E. Fischer

CSCI 1166 Discrete Mathematics for ComputingFebruary 14, 2018

Alice E. Fischer Laws of Logic. . . 1/34



ImplicationSummary

1 Properties of PropositionsLogical EquivalenceContradictions and Tautologies

2 Logical Equivalences

3 Practice with Boolean Operators and Algebra

4 ImplicationNecessary and Sufficient ConditionsInside out and Backwards

5 Summary




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Logical EquivalenceContradictions and Tautologies

Properties of Propositions





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Logical Equivalence: ≡Two propositions are logically equivalent (≡) if they have thesame outcomes under all input conditions.

We can determine whether two propositions are equivalent (≡) bycomparing the last column in the truth tables for the twostatements.

P Q P ∨ Q Q ∨ P

T T T TT F T TF T T TF F F F

For example, The column for P ∨ Q is the same as the column forQ ∨ P, so the two propositions are equivalent .




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Example: The Supervisor’s Critique

Do you remember John and his elevator design from last time?John’s supervisor looked at his logic and told him “It’s toocomplex, O is just the opposite of M”. Simplify! So John thoughta bit and added a column with a simpler formula to his truth table.

(O ∧ ∼M)∨

O M ∼O ∼M O ∧ ∼M ∼O ∧M (∼O ∧M) O ⊕MT T F F F F F FT F F T T F T TF T T F F T T TF F T T F F F F

Indeed, the boss was right. They are equivalent!




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Contradictions

A contradiction is a proposition that is always false, no matterwhat the input truth values are.

For example, this is a contradiction:An odd number, X, equals 2 * some even number, Y.

And this is a contradiction: P ∧ ∼PLet P be All dogs are black.All dogs are black and not (all dogs are black).

A contradiction can be symbolized by c.




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Tautologies

A tautology is a proposition that is always true, no matter whatthe input truth values are.

For example:Let Q be the proposition Today is Tuesday.

Then this is a tautology: Q ∨ ∼QToday is Tuesday or today is not Tuesday.

A tautology can be symbolized by t.




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Logical Equivalences

A set of rules for logical calculation can be proven from thedefinitions using truth tables. They will be presented in thefollowing slides.

These laws are analogs of the laws for algebra and for sets that youalready know. They are essential in making logical proofs andseveral are also important in programming.

The following slides define the laws for symbolic logic. Thecorresponding set-theory law(s) will be given at the bottom of eachslide.




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1. Commutative Laws

Both and and or are commutative, that is, the left and right sidescan be reversed without changing the meaning of the proposition.

P ∧ Q ≡ Q ∧ P

P ∨ Q ≡ Q ∨ P

This is important in programming.

Set theory: ∩ and ∪ are commutative. Given sets A and B,A ∩ B ≡ B ∩ AA ∪ B ≡ B ∪ A




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2. Associative Laws

Both and and or are associative operators, that is, you mayparenthesize the expression any way you wish without changing itsmeaning.

(P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R)

(P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R)


Set theory: ∩ and ∪ are associative. Given sets P, Q, and R,(P ∩ Q) ∩ R ≡ P ∩ (Q ∩ R)(P ∪ Q) ∪ R ≡ P ∪ (Q ∪ R)




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3. Distributive Laws

Both and and or can be distributed over the other operator.

P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)

P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)


Set theory: ∩ and ∪ can be distributed over each other.Given sets P, Q, and R,

P ∩ (Q ∪ R) ≡ (P ∩ Q) ∪ (P ∩ R)P ∪ (Q ∩ R) ≡ (P ∪ Q) ∩ (P ∪ R)




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4. Identity Laws

The identity laws are based on the definitions of and and or.

P ∧ t ≡ P

P ∨ c ≡ P

Set Theory: The set identity laws are based on definitions of theUniversal set and the empty set.Given set A,

A ∩ U ≡ AA ∪ ∅ ≡ A




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5. Negation Laws

The negation laws are based on the definitions of and, or, and not.

P ∨ ∼P ≡ t

P ∧ ∼P ≡ c

Set Theory: Complement LawsThe set complement laws are based on definitions of the Universalset, empty set, and complement.Given set A,

A ∪ Ac ≡ UA ∩ Ac ≡ ∅




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6. Double Negative Law

The double-negative law is based on the definitions of not.

∼ (∼P) ≡ P

Set Theory: Double Complement LawThis is based on the definitions of complement.Given set A,

(Ac)c ≡ A




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7. Idempotent Laws

and, or are idempotent, meaning that the trivial computationshave no effect and can be repeated without changing the meaningof the expression.

P ∧ P ≡ P

P ∨ P ≡ P

Set Theory: intersection and union are idempotent.Given set A,

A ∩ A ≡ AA ∪ A ≡ A




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8. Universal Bound Laws

A contradiction and anything is a contradiction.A tautology or anything is a tautology.

P ∧ c ≡ c

P ∨ t ≡ t

The shortcut evaluation of logical operators in C is based on these.

Set Theory: The empty set is as small as you can get and theuniversal set is as large as you can get. Given set A,

A ∩ ∅ ≡ ∅A ∪ U ≡ U




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9. DeMorgan’s Laws

To distribute not over and, change the and to or.To distribute not over or, change the or to and.

∼ (P ∧ Q) ≡ ∼P ∨ ∼Q∼ (P ∨ Q) ≡ ∼P ∧ ∼Q

These laws are very important in programming and in databases.

Set Theory:To distribute complement over ∩, change the ∩ to ∪.To distribute complement over ∪, change the ∪ to ∩.Given sets A,B,

(A ∩ B)c ≡ Ac ∪ Bc

(A ∪ B)c ≡ Ac ∩ Bc




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10. Absorption Laws

A truth table can be used to verify these absorptions.

P ∨ (P ∧ Q) ≡ P

P ∧ (P ∨ Q) ≡ P

These laws are important in proofs.

Set Theory:Given sets A,B,

A ∪ (A ∩ B) ≡ A That is, A ∪ a subset of A is still A.A ∩ (A ∪ B) ≡ A That is, A ∩ a superset of A is still A.




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11. Negations of t and c

The negation of a tautology is a contradiction and vice versa.

∼ t ≡ c

∼ c ≡ t

Set Theory:The complement of the Universal set is the empty set, and viceversa.

Uc ≡ ∅∅c ≡ U




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Practice

This section contains three logical expressions to simplified byapplying the rules, and one law to prove using truth tables.




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1. Proving DeMorgan’s Law

Construct a truth table to prove that ∼(P ∧ Q) ≡ ∼P ∨ ∼Q

P Q P ∧ Q ∼(P ∧ Q) ∼P ∼Q (∼P ∨ ∼Q)T T T F F F FT F F T F T TF T F T T F TF F F T T T T

Compare the contents of the fourth and seventh columns.They are the same, so the two expressions are equivalent.




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2. Two small messes to simplify.

(P∧ ∼ Q) ∨ (P ∧ Q) Initial expression.P ∧ (∼ Q ∨ Q) Use the distributive law, backwards.P ∧ (t) Use negation law.P Use identity law.

P ∧ (∼ Q ∨ P)(P∧ ∼ Q) ∨ (P ∧ P) Distributive law.(P∧ ∼ Q) ∨ P Idempotent lawP ∨ (P∧ ∼ Q) Commutative lawP Absorption law.




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3. A big mess.

(P ∧ (∼ (∼P ∨ Q))) ∨ (P ∧ Q) Initial expression(P ∧ (∼∼P) ∧ ∼Q)) ∨ (P ∧ Q) DeMorgan’s law.(P ∧ ( P ∧ ∼ Q)) ∨ (P ∧ Q) Double negative law.((P ∧ P) ∧ ∼Q) ∨ (P ∧ Q) Associative law.( P ∧ ∼Q) ∨ (P ∧ Q) Idempotent law.

P ∧ (∼Q ∨ Q) Distributive law, backwards.P ∧ (t) Negation law.P Identity law.




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Necessary and Sufficient ConditionsInside out and Backwards

Implication

→ and its Truth TableNecessary and Sufficient Conditions

Inside out and Backwards




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→ and its Truth Table

The symbol → is called if. . . then in the textbook.

It is more traditional to call it implies.

The proposition P → Q is read P implies Q or if P, then Q

We say that P is the premise and Q is the conclusion.

The truth table for P → Q is:

P Q P → Q ∼ P ∼ P ∨ Q

T T T F TT F F F FF T T T TF F T T T

It is also true that P → Q ≡ ∼ P ∨ Q




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Implication Can be Confusing.

Sometimes, but not always, P → Q means that P causes Q:I didn’t increment my loop variable so my program had aninfinite loop.

Sometimes P → Q is used to define a term:If a creature is bipedal, it has two feet.

The conclusion can be true independently of the premise: Ifthe moon is made of green cheese then UNH is a university.This is often called a vacuous truth.

The implication is true when both the premise and theconclusion are false. if 0 = 1 then 1 = 2.

A truth table is the most reliable way to figure out the correctinterpretation of an implication.




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If and Only If ↔

P → Q is a conditional statement that can be read if P then Q.

Now suppose that P → Q and also Q → P.

Then we write P ↔ Q, which can be read P if and only if Q.

This statement is called a biconditional.

P Q P → Q Q → P P ↔ Q

T T T T TT F F T FF T T F FF F T T T




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Necessary and Sufficient Conditions ↔

A sufficient condition is a premise or set of premises that areenough, without any more premises, to prove the conclusion.

If C is a sufficient condition for S , then C → S .

A necessary condition is a premise that must be true in orderfor the conclusion to be true.

If C is a necessary condition for S , then ∼C → ∼S .

If a condition C is both necessary and sufficient for S , thenC ↔ S .




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Inside Out and Backwards

There are four ways to “reverse” the proposition P → Q:

The contrapositive is ∼ Q → ∼P.

The converse is Q → P.

The inverse is ∼P → ∼Q.

The negation is ∼(P → Q), which is equivalent to P ∧ ∼Q.

A proposition and its contrapositive are equivalent.

A proposition is not equivalent to its converse, inverse, ornegation. For example,“If it is a bee, it can fly” is not ≡ to “If it can fly, it is a bee”.




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The four “opposites” of P → Q:

A conditional is equivalent to its contrapositive.

The inverse is equivalent to the converse and has the oppositetruth value of the proposition.

Conditional ConverseP → Q Q → P

Inverse Contrapositive∼P → ∼Q ∼Q → ∼P

P Q ∼P ∼Q P → Q ∼Q → ∼P Q → P ∼P → ∼QT T F F T T T TT F F T F F T TF T T F T T F FF F T T T T T T




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The Negation of P → Q:

When you need to negate an implication, it is usually best to workwith the form on the right.

∼(P → Q) ≡ ∼Q ∧ P

P Q ∼Q P → Q ∼(P → Q) ∼Q ∧ PT T F T F FT F T F T TF T F T F FF F T T F F




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Summary




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Summary of Terminology

A conjunction is two propositions joined by ∧.

A disjunction is two propositions joined by ∨.

A negation is ∼ applied to a proposition.

Two propositions are equivalent (≡) if the results in the lastcolumn of their truth tables are identical.

An implication is two propositions joined by →.

A biconditional is two propositions joined by ↔.




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More Terminology

A proposition that is always true is called a tautology.

A proposition that is always false is called a contradiction.

Given a proposition P → Q,

The contrapositive is ∼Q → ∼P (equivalent to proposition)

The converse is Q → P (equivalent to inverse)

The inverse is ∼P → ∼Q (equivalent to converse)

The negation is ∼(P → Q) (equivalent to P ∧ ∼Q)


equivalence and implication

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