section 1.5 implications. implication statements if cara has a piano lesson, then it is friday. if...
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Section 1.5
Implications
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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.
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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.
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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.
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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 …
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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 …
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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
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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?
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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
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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
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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.
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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.
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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”.
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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”.
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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.
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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
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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?
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Before next time you should…
Make sure that you have carefully read Section 1.5 and completed the homework assignment.