Equivalent expression

When you buy a car, the car can be new or old. Sales will say "of course not true this car is not a new car." This compound sentence has a one-sentence statement (This car is new) and two negation.

"Absolutely not true" "This car is not a new car"
~ ~ p

Does this mean a new car? To find out the truth, we can construct the truth table for logic expression ~ (~ p) and compare it with the p. Since there is only one single statement, then we only need 2 lines in the table. We will have a column p, ~ p and ~ (~ p). ~ p gives the value opposite p. Medium ~ (~ p) gives the value in contrast to ~ p. So the table would be as follows.

p	~p	~(~p)
B	S	B
S	B	S

Note that the third column of ~ (~ p) has a truth value is identical to p. If like this, then ~ (~ p) is said is equivalent to p. That is, meaning they are the same statement. An equivalent expression tends to be a symbolic expression that has an identical truth value to each other.

Expression p q read p is equivalent to q or p and q equivalent. From the above table we find that ~ (~ p) is equivalent to p, can we write p ~(~p).

EXAMPLE
Is the statement "If my landlord, who pays my property tax" and the statement "I am the owner of the house and I do not pay property tax" equivalent?

COMPLETION
We begin by giving a symbolic representation of the statements above.
P: My landlord
q: I pay property tax
p -> q: if I am the owner of the house, I pay property tax.
p ^ ~ p: I am the owner of the house and I do not pay property tax.
Truth table for this case will contain four lines. The table we will use as below.

p	q	~q	p^~q	p -> q
B	B
B	S
S	B
S	S

Now we just give the truth value corresponding to each column. Column ~ q, will we fill with the opposite truth value from column q. The next column of the contents of B in the second row and the S on the other line, since conjunction requires the truth value of B for each factor. Meanwhile in the last column, because the conditional is only false if p is true and q is false, then fill the second row with S and B on the other line.

p	q	~q	p^~q	p -> q
B	B	S	S	B
B	S	B	B	S
S	B	S	S	B
S	S	B	S	B

Because the truth values ​​in the column p ^ ~ q is not equal to p -> q, the two statements are not equivalent. Note that p ^ ~ q and p -> q has the truth values ​​of the exact opposite. When this happens in two statements, one compound statement is the negation of another compound statement. Consequently, p ^ ~ q is the negation of p -> q. This relationship can be expressed by p ^ ~ q ~ (p -> q). The negation of a conditional premise is equivalent to conjunction and negation conclusion.

The statement that looks different in reality may have the same purpose. When we have two statements are equivalent, we can replace each other without changing the meaning. Emotional factors that influence the selection of our statement in practice. Not on its meaning.