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Logic ChAPTER 3

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**Variations of the Conditional and Implications 3.4**

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**Variations of the conditional**

Variations of p → q Converse: Inverse: Contrapositive: p → q is logically equivalent to ~ q → ~ p q → p is logically equivalent to ~ p → ~ q

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**Examples of variations**

p: n is not an even number. q: n is not divisible by 2. Conditional: p → q If n is not an even number, then n is not divisible by 2. Converse: q → p If n is not divisible by 2, then n is not an even number. Inverse: ~ p → ~ q If n is an even number, then n is divisible by 2. Contrapositive: ~q → ~p If n is divisible by 2, then n is an even number.

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**p q T F p → q T F q → p T F ~p → ~ q T F ~q → ~p T F Equivalent**

Conditional Converse Inverse Contrapositive p q T F p → q T F q → p T F ~p → ~ q T F ~q → ~p T F

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**Conditional Equivalents**

Statement Equivalent forms if p, then q p → q p is sufficient for q q is necessary for p p only if q q if p

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**Biconditional Equivalents**

Statement Equivalent forms p if and only if q p ↔ q p is necessary and sufficient for q q is necessary and sufficient for p q if and only if p

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**Examples of variations**

Given: h: honk u: you love Ultimate Write the following in symbolic form. Honk if you love Ultimate. If you love Ultimate, honk. Honk only if you love Ultimate. A necessary condition for loving Ultimate is to honk. A sufficient condition for loving Ultimate is to honk To love Ultimate, it is sufficient and necessary that you honk. or

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**Tautologies and Contradictions**

A statement that is always true is called a tautology. A statement that is always false is called a contradiction.

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Example Show by means of a truth table that the statement p ↔ ~ p is a contradiction. p ~p p ↔ ~ p T F

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Implications The statement p is said to imply the statement q, p q, if and only if the conditional p → q is a tautology.

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Example Show that p q p → q (p → q) Λ p [(p → q) Λ p] → q T F END

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