Lesson LOGIC PROBLEM INVOLVING EQUIVALENCE USING DEMORGAN'S LAW WITH SOLUTION AND EXPLANATION

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Use De Morgan�s laws to determine whether the two statements are equivalent:
&#8764;p &#8744; &#8764;q, &#8764;(p &#8743; 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.

<a href = "http://www.bookrags.com/research/demorgans-laws-wom/" target = "_blank">http://www.bookrags.com/research/demorgans-laws-wom/</a>

Our truth table looks like this:
<pre>
    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
</pre>
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.