Use De Morganís laws to determine whether the two statements are equivalent:
∼p ∨ ∼q, ∼(p ∧ q)
You want to test for equivalency of the 2 statements.
the statements are:
(not p) or (not q)
not (p and q)
~ is the not symbol.
^ is the and symbol.
v is the or symbol.
<-> is the equivalency symbol.
If the statements are equivalent, then they will have the same truth table.
We'll construct a table and see what happens.
here's a reference you can look at. It discusses DeMorgan's Law.
Our truth table looks like this:
p q ~p ~q (p ^ q) (~p v ~q) (~(p ^ q)) (~p v ~q) <-> (~(p ^ q))
T T F F T F F T
T F F T F T T T
F T T F F T T T
F F T T F T T T
You start with the basic variables and create your truth table from that.
first column is p (truth table for p)
second column is q (truth table for q)
third column is ~p (truth table for not p - this is the negation of p)
fourth column is ~q (truth table for not q - this is the negation of q)
fifth column is (p ^ q) truth table for (p and q)
sixth column is (~p v ~q) truth table for (not p or not q)
seventh column is (~(p v q) truth table for (not (p and q) - this is the negation of (p and q)
eighth column is (~p v ~q) <-> (~(p ^ q))
this is the equivalency statement for (not p or not q) and (not (p and q)).
As seen from this table:
if p is true, then not p is false.
if p is false, then not p is true.
if q is true, then not q is false.
if q is false, then not q is true.
(p ^ q) is true only if p and q are both true. If they are mixed or both false, then (p ^ q) is false.
~(p^q) is the negation of (p ^ q).
if (p ^ q) is true, then ~(p ^ q) is false.
if (p ^ q) is false, then ~(p ^ q) is true.
(~p v ~q) is true if ~p is true or ~q is true.
only one of them is required to be true.
(~p v ~q) is false only if ~p is false and ~q is false. They both have to be false in order for (~p v ~q) to be false. If they are mixed or both true, then (~p v ~q) is true.
(~p or ~q) <-> (~(p ^ q)) is the equivalency statement.
it has the form of A if and only if B.
in this statement, A is equal to (~p or ~q) and B is equal to (~(p ^ q)).
since (~p or ~q) <-> (~(p ^ q)) is true under all conditions, then the 2 statements are equivalent.
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