# Discrete Math: Logic and Statements

### 36 terms by mobile_chris

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p -- > q

q --> p

~ p --> ~ q

~ q --> ~ p

### Logically Equivalent Conditions

Original (p --> q) === Contrapositive (~q --> ~p)
OR
Converse (q --> p) === Inverse (~p --> ~q)

p ^ ~ q

~ (p --> q)

### p -- > q (Orig) ===

~ q --> ~ p (Contrapositive)

### q --> p (Converse) ===

~ p --> ~ q (Inverse)

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

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"

p v q
~ p
... q [VALID]

### Transitivity

p --> q
q --> r
... p --> r [VALID]

p
... p v q

p ^ q
... p

p v q
p --> r
q --> r
... r [VALID]

p
q
... p ^ q

~ p --> C
... p

p --> q
q
... p [INVALID]

### Inverse Error

p --> q
~ p
... ~ q [INVALID]

~ p --> q

### DeMorgans Laws

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

~ p v 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))

Example: