Question

Are the two statements logically equivalent? Why or why not?

(p ∧ ~q) and ~(p ∨ q)

Answer 1

@Hero

Answer 2

i say yes, because they are true

Answer 3

logically equivalent means they give the same T or F when you put in the same p and q

to do this problem you list all possible values for p and for q
p q ~q p ^ ~q
F F T F ^ T = F
F T F F ^ F = F
T F T T ^ T = T
T T F T ^ F= F

now do the same for ~(p v q)

Answer 4

T
F
T
F

Answer 5

The first line says: let p be F and q be F
the ~q (not q) is the "flip" of q if q is F then ~q is T and vice versa

then for p ^ ~q you put in F for p and T for ~q and do the "and" operation
of course F and T is F
you do that for each row.

Answer 6

Also ~( p v q) can be translated to (~p v ~q)

Answer 7

yes that is the table for ~(p v q)
it does NOT match (p ∧ ~q)

so the expressions are not logically equivalent

as chaguanas points out (with a typo)
~(p v q) is logically equivalent to ~p ^ ~q and that is different from p ∧ ~q

Answer 8

so, you are saying that they are not logically equivalent

Answer 9

you get different answers, so they are not equivalent

if you tested
~(p v q) and ~p ^ ~q
you would see these two are logically equivalent

Answer 10

alright, i can see what you are saying now.

Answer 11

Thanks phi !

Answer 12

for ~(p v q) I get
T
F
F
F

if you then compute
(p ∧ ~q) = ~(p ∨ q)
F = T F
F = F T
T = F F
F = F T

if the = operation gave you T for every choice the two expression would be equivalent