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Contrapositive

~ q --> ~ p

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Logically Equivalent Conditions

Original (p --> q) === Contrapositive (~q --> ~p)

OR

Converse (q --> p) === Inverse (~p --> ~q)

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p -- > q (Orig) ===

~ q --> ~ p (Contrapositive)

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q --> p (Converse) ===

~ p --> ~ q (Inverse)

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~ q --> ~ p (Contrapositive) ===

p -- > q (Orig)

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~ p --> ~ q (Inverse) ===

q --> p (Converse)

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Conditional: If today is Easter, then tomorrow is Monday (write the Contrapositive)

Contrapositive: If tomorrow is not Monday, then today is not Easter

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Negation of p --> q

p ^ ~ q

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Conditional: If today is Easter, then tomorrow is Monday (write the Converse)

Converse: If tomorrow is Monday, then today is Easter.

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Conditional: If today is Easter, then tomorrow is Monday (write the Inverse)

Inverse: If today is not Easter, the tomorrow is not Monday.

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Conditional: If Tom is Ann's father, then Jim is her uncle and Sue is her aunt. (write Contrapositive)

Contrapositive: If either Jim is not Ann's uncle or Sue is not her aunt, then Tom is not her father

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Conditional: If X, then Y AND Z

Contrapositive: If EITHER not Y OR not Z, then not X

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Conditional: If X, then Y OR Z

Contrapositive: If not Y AND Z, then not X

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Modus Ponens

p --> q

p

... q [VALID]

modus ponendo ponens: "the way that affirms by affirming"

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Modus Tollens

p --> q

~ q

... ~ p [VALID]

modus tollendo tollens: "the way that denies by denying"

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Elimination

p v q

~ p

... q [VALID]

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Transitivity

p --> q

q --> r

... p --> r [VALID]

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Generalization

p

... p v q

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Specialization

p ^ q

... p

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Proof by Division into cases

p v q

p --> r

q --> r

... r [VALID]

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Conjunction

p

q

... p ^ q

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Contradiction

~ p --> C

... p

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Converse Error

p --> q

q

... p [INVALID]

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Inverse Error

p --> q

~ p

... ~ q [INVALID]

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DeMorgans Laws

~(p ^ q) === ~p v ~q

~(p v q) === ~p ^ ~q

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p <--> q ===

(p --> q) ^ (q --> p)

(p ^ q) v (~p ^ ~q)

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Vacuously True

Hypothesis is False

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Implication is False

ONLY when hyp (p) is T and conc (q) is F

(p(T) --> q(F))

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