introduction to set theory and logic (intoset) · introduction the rules of logic give precise...

Propositions and Logical Operators Tautology, Contradiction and Contingency Rules of Replacement

Introduction to Set theory and Logic(INTOSET)

Fundamentals and Algebra of Logic

Francis Joseph Campena,PhDMathematics and Statistics Department

De La Salle University-Manila


Introduction

The rules of logic give precise meaning to mathematicalstatements. These rules are used to distinguish between validand invalid mathematical arguments.Our discussion begins with an introduction to the basic buildingblocks of logic-propositions.


Propositions

DefinitionA proposition is a declarative sentence (that is, a sentencethat declares a fact) that is either true or false, but not both.

ExampleAll the following declarative sentences are propositions.

1 Washington, D.C., is the capital of the United States ofAmerica.

2 Manila is in the island of Visayas.3 1 + 1 = 2.4 2 + 2 = 3.


Propositions

ExampleConsider the following sentences.

1 What time is it?2 Read this carefully.3 x + 1 = 2.4 x + y = z.

The first two sentences are not propositions since they are notdeclarative sentences. Sentence 3 and 4 are not propositionssince they are neither true nor false. However, they can bemade into a proposition if we assign values to the variables.


Propositional Variables

We use letters to denote propositional variables (orstatement variables), that is, variables that representpropositions, just as letters are used to denote numericalvariables.The truth value of a proposition is true, denoted by T, if it isa true proposition, and the truth value of a proposition isfalse, denoted by F, if it is a false proposition.Many mathematical statements are constructed bycombining one or more propositions. New propositions,called compound propositions, are formed from existingpropositions using logical operators.


Logical Operators

Some of the logical operators that we are going to look into areas follows:

1 Negation (¬)

2 Conjunction (∧)

3 Disjunction (∨)


Logical Operator: NEGATION

DefinitionLet p be a proposition. The negation of p, denoted by ¬p (alsodenoted by p̄), is the statement

"It is not the case that p."

The proposition ¬p is read "not p." The truth value of thenegation of p, ¬p, is the opposite of the truth value of p

ExampleConsider the proposition P: Lynne’s PC runs Linux. Thenegation of P, ¬P is "It is not the case that Lynne’s PC runsLinux." Or we just simply say that Lynne’s PC does not runLinux.


Logical Operator: CONJUNCTION

DefinitionLet p and q be propositions. The conjunction of p and q,denoted by p ∧ q is the proposition ”p and q.” The conjunctionp ∧ q is true when both p and q are true and is false otherwise.

Note that in logic the word "but" sometimes is used instead of"and" in a conjunction. For example, the statement "The sun isshining, but it is raining" is another way of saying "The sun isshining and it is raining." (In natural language, there is a subtledifference in meaning between "and" and "but"; we will not beconcerned with this nuance here.)


Logical Operator: Disjunction

DefinitionLet p and q be propositions. The disjunction of p and q,denoted by p ∨ q, is the proposition "p or q." The disjunctionp ∨ q is false when both p and q are false and is true otherwise.

RemarkThe use of the connective or in a disjunction corresponds toone of the two ways the word or is used in English, namely, asan inclusive or. A disjunction is true when at least one of thetwo propositions is true.


Logical Operator: Disjunction

RemarkFor instance, the inclusive or is being used in the statement

“Students who have taken calculus or computer science cantake this class."

Here, we mean that students who have taken both calculus andcomputer science can take the class, as well as the studentswho have taken only one of the two subjects. On the otherhand, we are using the exclusive or when we say

“Students who have taken calculus or computer science, butnot both, can enroll in this class."


Truth Tables

The following are the truth tables for the negation of aproposition, conjunction of two propositions and the disjunctionof two propositions.


Conditional Statements or Implications

DefinitionLet p and q be propositions. The conditional statement p → qis the proposition "if p, then q."

The conditional statement p → q is false when p is trueand q is false, and true otherwise.In the conditional statement p → q, p is called thehypothesis (or antecedent or premise) and q is calledthe conclusion (or consequence).



The statement p → q is called a conditional statementbecause p → q asserts that q is true on the condition thatp holds.A conditional statement is also called an implication.Note that the statement p → q is true when both p and qare true and when p is false (no matter what truth value qhas).



The following are some ways of expressing a conditionalstatements:

"if p, then q""p implies q""p only if q"

"p is sufficient for q""q whenever, p""q is necessary for p"

The following is the truth table for the conditional statementp → q.


Biconditional

DefinitionLet p and q be propositions. The biconditional statementp ⇔ q is the proposition "p if and only if q."

The biconditional statement p ⇔ q is true when p and qhave the same truth values, and is false otherwise.Some common ways to express p ⇔ q are: "p isnecessary and sufficient for q", "p iff q" or "if p then q, andconversely"


Remarks

A. The words "both" goes with "and" and "either" goes with "or"have parenthetical meanings. So, we have the following:

1 "Both P or Q and R" is represented by (P ∨Q) ∧ R.2 "P or both Q and R" is represented by P ∨ (Q ∧ R).3 "Either P and Q or R" is represented by (P ∧Q) ∨ R.4 "P and either Q or R" is represented by P ∧ (Q ∨ R).

B. The expression "neither P nor Q" is the same as "not eitherP or Q" and is therefore denoted by ¬(P ∨Q).

C. The truth values of a propositional form can be shownthrough a truth table. A truth table for a propositional formwhich has n propositional variables as components has 2n

number of rows.


EXERCISES

Write the following sentences in symbolic form and construct itscorresponding truth table. Explicitly define the propositionalvariables you intend to use.1. If Ruteza learned to read and write well, she got a good job

and made lots of money.2. If we don’t control the money supply and break the power of

OPEC, we won’t control inflation.3. Either we control the money supply and break the power of

OPEC, or we won’t control inflation and the economy willcollapse.

4. Either the Russians and Americans both reduce theirnuclear arsenals or neither will.


EXERCISES

Construct the truth table of the compound propositions1 P ∧ ¬P2 P ∨ ¬P3 P ∨ ¬Q → Q4 (P ∨ ¬Q)→ (P ∧Q)

5 (P → Q)⇔ (¬Q → ¬P)


Tautology, Contradiction and Contingency

An important type of step used in a mathematical argument isthe replacement of a statement with another statement with thesame truth value. Because of this, methods that producepropositions with the same truth value as a given compoundproposition are used extensively in the construction ofmathematical arguments. Note that we will use the term"compound proposition" to refer to an expression formed frompropositional variables using logical operators, such as p ∧ q.


Tautology, Contradiction and Contingency

DefinitionA compound proposition that is always true, no matterwhat the truth values of the propositional variables thatoccur in it, is called a tautology.A compound proposition that is always false is called acontradiction or absurdity.A compound proposition that is neither a tautology nor acontradiction is called a contingency.

ExampleWe can construct examples of tautologies and contradictionsusing just one propositional variable. Consider the truth tablesof P ∨ ¬P and P ∧ ¬P. Because P ∨ ¬P is always true, it is atautology. Because P ∧ ¬P is always false, it is a contradiction


Logical Equivalence

Compound propositions that have the same truth values in allpossible cases are called logically equivalent.

DefinitionThe compound propositions p and q are called logicallyequivalent if p ⇔ q is a tautology. The notation p ≡ q denotesthat p and q are logically equivalent.

Note that in the above Truth Table, the compound propositions¬(p ∨ q) and ¬p ∧ ¬q are logically equivalent.


Exercises

1. Show that p → q and ¬p ∨ q are logically equivalent.2. Use the short version of the truth table to show that the

following are logically equivalent: p ∨ (q ∧ r) and(p ∨ q) ∧ (p ∨ r).


Rules of Replacement

The following are some important equivalences. SupposeP,Q,R are propositions and where 1 denotes a compoundproposition that is always True and 0 denotes a compoundproposition that is always False.

Identity(P ∨ 1)⇔ 1(P ∨ 0)⇔ P(P ∧ 1)⇔ P(P ∧ 0)⇔ 1(P ∨ ¬P)⇔ 1(P ∧ ¬P)⇔ 0¬0⇔ 1¬1⇔ 0

Idempotent LawsP ⇔ (P ∨Q)P ⇔ (P ∧Q)

Commutative Laws(P ∨Q)⇔ (Q ∨ P)(P ∧Q)⇔ (Q ∧ P)

Associative Laws[(P ∨Q) ∨ R]⇔[P ∨ (Q ∨ R)[(P ∧Q) ∧ R]⇔[P ∧ (Q ∧ R)]

De Morgan’s Laws¬(P ∨Q)⇔ (¬P ∧ ¬Q)¬(P ∧Q)⇔ (¬P ∨ ¬Q)


Rules of Replacement

Distributive Laws[P ∨ (Q ∧ R)]⇔ [(P ∨Q) ∧ (P ∨ R)][P ∧ (Q ∨ R)]⇔ [(P ∧Q) ∨ (P ∧ R)]

Material Equivalence[P ⇔ Q]⇔ [(P ⇒ Q) ∧ (Q ⇒ P)][P ⇔ Q]⇔ [(P ∧Q) ∨ (¬P ∧ ¬Q)]

Involution/Double Negation LawP ⇔ ¬¬P

Material Implication(P ⇒ Q)⇔ (¬P ∨Q)

Exportation[(P ∧Q)⇒ R]⇔ [P ⇒ (Q ⇒ R)]

Absurdity[(P ⇒ Q) ∧ (P ⇒ ¬Q)]⇔ ¬P

Contrapositive(P ⇒ Q)⇔ (¬Q ⇒ ¬P)


EXERCISES

For each of the following arguments, state the Rule ofReplacement by which each conclusion (the propositional formpreceded by ∴ ) follows from its premise.

1(¬A⇒ B) ∧ (C ∨ ¬D)∴ (¬A⇒ B) ∧ (¬D ∨ C)

2(I ⇒ J) ∨ (¬K ⇒ ¬L)∴ (I ⇒ ¬J) ∨ (L⇒ K )

3M ⇒ ¬(N ∨ ¬O)

∴ M ⇒ (¬N ∧ ¬¬O)

4(¬E ∨ F ) ∧ (G ∨ ¬H)∴ (E ⇒ F ) ∧ (G ∨ ¬H)


EXERCISES

For each of the following arguments, state the Rule ofReplacement by which each conclusion (the propositional formpreceded by ∴ ) follows from its premise.

1(U ∨ V ) ∧ (W ∨ X )

∴ [(U ∨ V ) ∧ X ] ∨ [(U ∨ V ) ∧ X ]

2[E ⇒ (F ∨G)] ∨ [E ⇒ (F ∨G)]

∴ E ⇒ (F ∨G)

3¬M ⇒ {N ⇒ [¬(O ∧ P)⇒ ¬Q]}∴ ¬M ⇒ {[N ∧ ¬(O ∧ P)]⇒ ¬Q}

4[H ∧ (I ∧ J)]⇒ (K ⇔ ¬L)

∴ H ⇒ [(I ∧ J)⇒ (K ⇔ ¬L)]

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