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Logic Day Two

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Biconditional The Biconditional is a compound statement that combines 2 conditionals and the connector “and” (p q) (q p). That is (p implies q) and (q implies p). It is written as p q. A biconditional is called an if and only if statement.

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**A biconditional statement is only true when both p and q have the same truth values.**

p q T F

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**Two statements are logically equivalent if each statement has the same truth values.**

To show this we us the logical connector if and only if. In symbolic form if and only if is represented by the symbol: Example: Prove that pq is logically equivalent to ~pq

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p q ~p ~pq pq (p q) (~p q) T F

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**Inverse, Converse and Contrapositives**

In conducting an argument the conditional statement is the one we use most often. In order to use them correctly we must understand their different forms and how they are related.

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The Inverse The inverse is formed by negating both the hypothesis and the conclusion. Example If a number is divisible by 4 then it is also divisible by 2. The inverse: If a number is not divisible by 4 then it is not divisible by 2.

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**A true conditional can have a false inverse**

(as seen in the last example). Sometimes a false conditional can have a true inverse. Sometimes both the statement and the inverse have the same truth values.

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**In symbolic form The inverse written in symbolic form Examples**

p q is the statement ~p ~q is the inverse. or ~p q is the statement then p ~q is the inverse.

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The Converse The converse is formed by interchanging both the hypothesis and the conclusion. Example If a polygon has four sides then it is a quadrilateral. (the statement) If a polygon is a quadrilateral then it has four sides. (the converse)

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Like the inverse a conditional statement can have a false converse, a true converse or they can have the same truth values. In symbolical form: pq is the statement q p is the converse.

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The Contrapositive. The contrapositive is formed by doing an inverse followed by a converse or a converse followed by an inverse.

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**The conditional statement and it’s contrapositive are equivalent statements.**

pq is logically equivalent to ~q ~p

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p q ~p ~q pq ~q ~p T F

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**Homework In the text book Pg. 67-68 4-28 Evens Pg. 73-74**

3, 7, 8, 10, 12,17a,

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