Section 1.5

Implications

Implication Statements

If Cara has a piano lesson, then it is Friday.

If it is raining, then I need to remember my umbrella.

If your MAT225 average is 95%, then you will receive an A in the course.

Definitions:

A statement of the form “if p is true, then q is true” is called an implication. We write pq.

In the statement “if p, then q,” we call p the hypothesis and q the conclusion.

p and q can indicate either propositions or predicates.

Mathematical statements

Which of the following mathematical statements seem to state “universal truths” about the positive integers?

1. If n is odd, then n2 – 1 is evenly divisible by 8.2. If n is evenly divisible by 3, then n2 + n is evenly

divisible by 4.3. If n ends in the digit “2”, then n is divisible by 2.4. If n ends in the digit “3”, then n is divisible by 3.5. If n or m is odd, then n + m is odd.6. If n or m is even, then n × m is even.

Truth of implicational statements

Using the false statements below as models, complete the sentence that follows in your own words.

If n is evenly divisible by 3, then n2 + n is evenly divisible by 4.

If n ends in the digit “3”, then n is divisible by 3. If n or m is odd, then n + m is odd.

To show that a predicate of the form P(x) Q(x) is false on the domain D, we must …

Truth of implicational statements

Using the true statements below as models, complete the sentence that follows in your own words.

If n is odd, then n2 – 1 is evenly divisible by 8. If n or m is even, then n × m is even. If n ends in the digit “2”, then n is divisible by 2.

A predicate of the form P(x) Q(x) is true for all elements of the domain D if …

Truth tables for if,then statements

Example. Here is the truth table for the statement, “p q”

p q p q

T T T

T F F

F T T

F F T

Making sense of the truth table for pqLet p := “your final average is at least

60%”Let q := “you pass MAT225”

If I make the statement pq, when am I a truthteller? When am I a liar?

Truth tables for if,then statements

Practice. Complete the truth table for the compound statement, “(¬q) (¬p)”

p q ¬q ¬p (¬q) (¬p)

T T F F

T F T F

F T F T

F F T T

Truth tables for if,then statements

Practice. Give the truth table for the compound statement, “(p (q p)) q”

p q q p p (q p) (p (q p)) q

T T

T F

F T

F F

Implication in English

The English construction “If property 1 holds, then property 2 holds” states a relationship between properties 1 and 2. In the investigation of this type of relationship, there is some standard terminology you should know:

The statements p q and q p are called converses of each other. It is possible but not unusual for only one statement in such a pair to be true.

The statements p q and (¬q) (¬p) are called contrapositives of each other. These statements are always logically equivalent.

The statements p q and (¬p) (¬q) are called inverses of each other. It is possible but not unusual for only one statement in such a pair to be true.

Examples

Form the converse, inverse, and contrapositive of each of the following statements:

1.If n ends in a “2”, then n is divisible by 2.

2.If n ends in a “3”, then n is divisible by 3.

3.If n ends in an even digit, then n is even.

Examples

If n ends in a “2”, then n is divisible by 2.

Converse: If n is divisible by 2, then n ends in a “2”.

Inverse: If n does not end in a “2”, then n is not divisible by 2.

Contrapostive: If n is not divisible by 2, then n does not end in a “2”.

Examples

If n ends in a “3”, then n is divisible by 3.

Converse: If n is divisible by 3, then n ends in a “3”.

Inverse: If n does not end in a “3”, then n is not divisible by 3.

Contrapostive: If n is not divisible by 3, then n does not end in a “3”.

Examples

If n ends in an even digit, then n is even.

Converse: If n is even, then n ends in an even digit.

Inverse: If n does not end in an even digit, then n is not even.

Contrapostive: If n is not even, then n does not end in an even digit.

Implication in predicate logic

The majority of mathematical statements can be written in the form

The negation of this statement is the formal statement

)()(, xQxPDx

)()(, xQxPDx

Examples in English

If a student at Shippensburg majors in math, then that student takes Discrete Math.

1. Write this using formal predicate logic.

2. What is the negation of this sentence?

Before next time you should…

Make sure that you have carefully read Section 1.5 and completed the homework assignment.