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Logic Day Two

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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. Biconditional

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A biconditional statement is only true when both p and q have the same truth values. pq p q TTT TFF FTF FFT

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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 p q is logically equivalent to ~p q

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pq~p ~p qp q(p q) (~p q) TTFTTT TFFFFT FTTTTT FFTTTT

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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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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. In symbolic form

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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: p q 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 its contrapositive are equivalent statements. p q is logically equivalent to ~q ~p

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pq~p~q p q~q ~p TTFFTT TFFTFF FTTFTT FFTTTT

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Homework In the text book Pg Evens Pg , 7, 8, 10, 12,17a,

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