36 terms

# Discrete Math: Logic and Statements

###### PLAY
Implication
p -- > q
Converse
q --> p
Inverse
~ p --> ~ q
Contrapositive
~ q --> ~ p
Logically Equivalent Conditions
Original (p --> q) === Contrapositive (~q --> ~p)
OR
Converse (q --> p) === Inverse (~p --> ~q)
~ (p --> q) ===
p ^ ~ q
p ^ ~ q ===
~ (p --> q)
p -- > q (Orig) ===
~ q --> ~ p (Contrapositive)
q --> p (Converse) ===
~ p --> ~ q (Inverse)
~ q --> ~ p (Contrapositive) ===
p -- > q (Orig)
~ p --> ~ q (Inverse) ===
q --> p (Converse)
Conditional: If today is Easter, then tomorrow is Monday (write the Contrapositive)
Contrapositive: If tomorrow is not Monday, then today is not Easter
Negation of p --> q
p ^ ~ q
Conditional: If today is Easter, then tomorrow is Monday (write the Converse)
Converse: If tomorrow is Monday, then today is Easter.
Conditional: If today is Easter, then tomorrow is Monday (write the Inverse)
Inverse: If today is not Easter, the tomorrow is not Monday.
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
Conditional: If X, then Y AND Z
Contrapositive: If EITHER not Y OR not Z, then not X
Conditional: If X, then Y OR Z
Contrapositive: If not Y AND Z, then not X
Modus Ponens
p --> q
p
... q [VALID]
modus ponendo ponens: "the way that affirms by affirming"
Modus Tollens
p --> q
~ q
... ~ p [VALID]
modus tollendo tollens: "the way that denies by denying"
Elimination
p v q
~ p
... q [VALID]
Transitivity
p --> q
q --> r
... p --> r [VALID]
Generalization
p
... p v q
Specialization
p ^ q
... p
Proof by Division into cases
p v q
p --> r
q --> r
... r [VALID]
Conjunction
p
q
... p ^ q
~ p --> C
... p
Converse Error
p --> q
q
... p [INVALID]
Inverse Error
p --> q
~ p
... ~ q [INVALID]
p OR q ===
~ p --> q
DeMorgans Laws
~(p ^ q) === ~p v ~q
~(p v q) === ~p ^ ~q
p --> q ===
~ p v q
p --> q ===
~q --> ~p
p <--> q ===
(p --> q) ^ (q --> p)
(p ^ q) v (~p ^ ~q)
Vacuously True
Hypothesis is False
Implication is False
ONLY when hyp (p) is T and conc (q) is F
(p(T) --> q(F))