Chapter 2 Logic

• 2.1 Statements• 2.2 The Negation of a Statement• 2.3 The Disjunction and Conjunction of Statements• 2.4 The Implication• 2.5 More on Implications• 2.6 The Biconditional• 2.7 Tautologies and Contradictions• 2.8 Logical Equivalence• 2.9 Some Fundamental Properties of Logical Equivalence• 2.10 Quantifies Statements

2.1 Statements

A statement is a declarative sentence or assertion that is true or false (but not both).

Example: The integer 2 is even. The integer -4 is postive.

Every statement has a true value: true (T), or false (F).

We often use P, Q and R to denote statements, or perhaps P1, P2, …, Pn if several statements are involved.

For example: P1: The integer 2 is even.

P2: The integer - 4 is positive.

P1 and P2 are both statements, where P1 has truth value T and P2 has truth value F.

Sentences That are Not Statements

The following sentences are NOT statements:• Commands:

Example: Divide 4x by the number 2

• Questions:

Example: Is the integer 3 even?

• Exclamatory:

Example: What a difficult problem!

Open Sentence

Example: Is “the number x is an integer” a statement?

An open sentence is a declarative sentence that satisfies the following properties:

• It contains one or more variables, each variable representing a value in some prescribed set, called the domain of the variable;

• It becomes a statement when values from their respective domain are substituted for these variables.

Example: The open sentence “x+4=10” is an open sentence. It is true only when x=6.

Representation

An open sentence that contains a variable x is represented by P(x), Q(x), or R(x).

If P(x) is an open sentence, where the domain of x is S, then we say P(x) is an open sentence over the domain S. Here P(x) is a statement for each x S.

Example: the open statement P(x): 0<x-3<5 over the domain Z is a true statement when x {4, 5, 6, 7}, and it is a false statement otherwise.

Truth Table

The possible true values of a statement are often given in a table, called a true table.

Example: True table for P is P , the truth table for Q is Q T T F FThe truth table for two statements P and Q is P Q T T T F F T F FWhat is the truth table for three statements P, Q and R?In general, a truth table involving n statements contains 2n possible

combinations of truth values for these statements.

Section 2.2 The Negation of a Statement

We are interested in investigating the truth or falseness of new statements that can be produced from one or more given statements by performing certain operations on them.

The negation of a statement P is the statement: not P, and is denoted by P. Although P could always be expressed as “It is not the case that P”.

Example: P: The integer 4 is even. P: The integer 4 is not even, or the integer

is odd.

P: The number e is a rational number. P: The number e is not a rational number, or the integer e is a irrational number.

Truth Table for Negation

The negation of a true statement is always false, and the negation of a false statement is always true. Here is the true table for P (in terms of the possible truth values of P).

P P

T F

F T

Section 2.3 The Disjunction and Conjunction of Statements

The disjunction of the statements P and Q is the statement: P or Q, and is denoted by PQ.

The disjunction is true if at least one of P and Q is true; otherwise, PQ is false. Therefore, PQ is true if exactly one of P and Q is true or both P and Q are true.

Example: P1: The integer 4 is even. P2: The integer 10 is prime.

P1 P2: Either 4 is even or 10 is prime.

It is true since at least one of P1 and P2 is true.

Truth Table for Disjunction

For two statements P and Q, the truth table for PQ is

P Q PQT T T

T F T

F T T

F F F

Conjunction of Statements

The conjunction of the statement P and Q is the statement: P and Q, and is denoted by PQ.

Example: P1: The integer 4 is even. P2: The integer 10 is prime.

P1 P2: 4 is even and 10 is prime. It is false since at P2 is false.

Truth Table for Conjunction

The conjunction PQ is true only when both P and Q are true; otherwise, PQ is false.

For two statements P and Q, the truth table for PQ is

P Q P QT T T

T F F

F T F

F F F

Section 2.4 The Implication

For statements P and Q, the implication is the statement: If P, then Q, and is denoted by PQ. In addition to the wording “if P, then Q”, we also express PQ in words as “P implies Q”.

Example: P1: The integer 4 is even. P2: The integer 10 is prime.

P1 P2: If 4 is even, then 10 is prime. (It is false.)

P2 P1: If 10 is prime, then 4 is even. (It is true, however.)

Example: The teacher promised If you earn an A on the final exam (P), then you receive an A for your

final grade (Q). Note: If P is false and Q is true, P Q is still true since you may get an

A from other reasons.If P is false and Q is false, P Q is still true since the teacher promised

nothing if you didn’t get an A on the final exam.

Truth Table for Implication

Note that the only situation for PQ is false is when P is true and Q is false, and PQ is true otherwise.

For two statements P and Q, the truth table for PQ is

P Q PQ

T T T

T F F

F T T

F F T

Expression of the Implication

The implication PQ can be expressed a the following:• If P, then Q.• Q if P.• P implies Q.

• P only if Q.• P is sufficient for Q.• Q is necessary for P.

Section 2.5 More on Implications

Just as new statements can be formed from statements P and Q by negation, disjunction, conjunction, or implication, new open statements can be constructed from open statements in the same manner.

Example

Example: Consider the open sentences P1: x=-2 and P2: |x|=2,

where x R. We can form the following open sentences. P1(x): x-2.

• P1 (x) P2(x): x=-2 or |x|=2.

• P1 (x) P2(x): x=-2 and |x|=2.

• P1(x) P2(x): If x=-2, then |x|=2.

For each given value of x R, the truth value of each resulting statement can be determined.

P1(3) is true;

P1 (2) P2(2) is true;

P1 (2) P2(2) is false;

P1(x) P2(x) is true for all x R. Why?

Example

In general, the sentence P in the implication PQ is referred to as the hypothesis or premise of PQ, while Q is called the conclusion of PQ.

Example: Let S={2, 3, 5} and let P(n): n-1 is even, and Q(n): n+3 is odd.

be open sentences over the domain S. Determine the truth or falseness of the implication for each n S.

Solution: P(2)Q(2) is ?

P(3)Q(3) is ?

Section 2.6 The Biconditional

For statement (or open sentences) P and Q, the implication P Q is called the converse of P Q.

Example: P1: The integer 4 is even. P2: The integer 10 is prime.

The converse of the implication P1 P2: If 4 is even, then 10 is prime. Is the implication P2 P1: If 10 is prime, then 4 is even.

For statements (or open sentences) P and Q, the conjunction

(P Q) (Q P)

is called the biconditional of P and Q, is denoted by P Q.

Truth Table for the Biconditional

The truth table for the biconditional PQ is P Q PQ

T T TT F FF T FF F T

Why?We see that PQ is true precisely when P and Q have the same truth

values.The biconditional PQ is often stated as P is equivalent to Q or P is and only if Qor P is necessary and sufficient condition for Q

Example

For statements P and Q, it then follows that the biconditional “P if and only if Q” is true only when P and Q have the same truth values.

Example: Consider the open sentences P1: x=-2 and P2: |x|=2,

where x R. P1(x) P2(x): If x=-2, then |x|=2.

Then P1(x) P2(x) is false when x=2, and is true for all other real numbers x. Why?

Example:

Let P(x, y): |x|=|y| and Q(x, y):x=y, where (x, y) {(6, 6), (2, -2)}.

Solution: P(6, 6) is true and Q(6, 6) is true, so P(6, 6) Q(6, 6) is true;

P(2, -2) is true and Q(2, -2) is false, so P(2, -2) Q(2, -2) is false.

Section 2.7 Tautologies and Contradictions

The symbols , , , , and are sometimes referred to as logical connectives. A compound statement is a statement composed of one or more given statements, and at least one logical connective.

Example: P (P) is a compound statement. Furthermore, it is true regardless of the truth value of P.

A compound statement S is called a tautology if it is true for all possible combinations of truth values of the component statements that comprise S.

Example: P (P), and (Q) (P Q) are both tautologies. Verify them.Example: Let P: 4 is even, then P (P): 4 is even or 4 is not even is

always true.

Contradictions

On the other hand, a compound statement S is called a contradiction if it is false for all possible combination of truth values of the component statements that are used to form S.

Example: P (P) and (P Q) (Q(P)) are both contradictions. Verify them.

Indeed, if a compound statement is a tautology, then its negation S is a contradiction.

2.8 Logical Equivalence

Give a truth table for the two statements: P Q and (P) Q.

If two (compound) statements R and S have the same truth values for combinations of truth values of their component statements, then we say that R and S are logically equivalent and indicate this by writing R S.

Example: P Q (P) Q.

Logically Equivalence and Tautology

Suppose R and S are logically equivalent compound statements. Then by the definition, the biconiditional RS is true for all possible combinations of truth values of their component statements and hence RS is a tautology.

Conversely, if RS is a tautology, then R and S are logically equivalent.

Theorem

Let P and Q be two statements. Then P Q and (P) Q. are logically equivalent.

PQ can be expressed as “P if and only if Q”, and “P is necessary and sufficient for Q”.

Section 2.9 Some Fundamental Properties of Logical Equivalence

Verify the following by means of a truth table.• P (P)

Theorem For statements P, Q and R,1. Commutative Laws

(a) P Q Q P(b) P Q Q P

2. Associative Laws(a) P (Q R) (P Q) R(b) P (Q R) (P Q) R

3. Distributive Laws(a) P (Q R) (P Q) (P R)(b) P (Q R) (P Q) (P R)

Theorem

4. De Morgan’s Laws(a)(P Q) ( P) ( Q)

(b)(P Q) ( P) ( Q)

Each part of the theorem can be verifies by means of a truth table.

The laws given here, together with other known logical equivalences, can be used to good advantage at times to prove other logical equivalences (without introducing a truth table).

Theorem: For statements P and Q,(a) (P Q) (P) ( Q)

(b)(P Q) ((P) ( Q)) ((Q) ( P))

Section 2.10 Quantified Statements

There is a method that an open sentence can be converted into a statement, called quantification.

Let P(x) be an open sentence over a domain S. Adding the phrase “For every x S” to P(x) produces a statement called a quantified statement.

The phrase “for every” is referred to as the universal quantifier and is denoted by the symbol . Other ways to express the universal quantifier are “for each” and “for all”.

Universal Quantifier

This quantified statement is expressed in symbols by

x S, P(x) (1)

and is expressed in words by

For every x S, P(x).

The quantified statement (1) is true if P(x) is true for every x S; while the quantified statement (1) is false if P(x) is false for at least one element x S.

Examples

Note that the quantified statement x S, P(x) can be expressed as

If x S, then P(x).

Example: x R, x2≥0 can be expressed as

For every real number, x2≥0.

or

If x is a real number, then x2≥0.

Notices that x R, x2≥0 is true since x2≥0 is true for every real number x.

Example: x R, x2>0 is false when x=0.

Existential Quantifier

Another way to convert an open sentence P(x) over a domain S into a statement is by a quantifier called an existential quantifier.

Each of the phrases “there exists”, “there is”, “for some”, and for “at least one” is referred to as an existential quantifier and is denoted by the symbol .

This quantified statement is expressed in symbols by

x S, P(x) (1)

and is expressed in words by

There exists x S such that P(x).

The quantified statement (1) is true if P(x) is true for at least one element x S; while the quantified statement (1) is false if P(x) is false for all x S.

Examples

Example: x R, x2>0 is true, but x R, x2<0 false.

Generally, if we are considering an open sentence P(x) over a domain S, then

(x S, P(x)) x S, P(x).

And

( x S, P(x)) x S, P(x).

Example: (x R, x2≤0) x R, x2>0).

Example: ( x R, 3x=12) x R, 3x≠12.

Example

The statement

There exists a real number x such that x2=3.• Express this using a quantifier.

x R, x2=3.

• Is it true or false?

True.• Express in word the negation of the statement.

For every real number x, x2≠3.

• Express the negation of the statement in symbols.

x R, x2≠3.

False.

Generalized Case

Let P(x, y) be an open sentence, where the domain of the variable x in S and domain of the variable y is T. The quantified statement

For all x S and y T, P(x, y).

Can be expressed symbolically as

x S, y T, P(x, y). (1)

The negation of the statement

(x S, y T, P(x, y)) x S, y T, P(x, y).