Proving Biconditional Equivalence: A Step-by-Step Guide

  • Thread starter maxsthekat
  • Start date
In summary, the conversation discusses the process of proving the equivalence of the expressions !(p <-> q) and (p <-> !q). The individual steps for both the left and right hand sides are outlined and the use of DeMorgan's theorem is mentioned. The conversation concludes with a mention of using truth tables or developing the expressions using distributivity of *.
  • #1
maxsthekat
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Homework Statement


I have to prove that !(p <-> q) = (p <-> !q)


2. The attempt at a solution
I started by trying to just work out what each side of the equation was. So, starting with the left hand side
!(p <-> q) = !((p->q) * (q->p)) (biconditional law)
= !((!p + q) * (!q + p)) (implication law)
= !(!p + q) + !(!q + p) (DeMorgan's theorem)
= (p * !q) + (q * !p) (DeMorgan's theorem)

Then, for the right hand side, I tried doing similarly:
(p <-> !q) = (p -> !q) * (!q -> p) (biconditional law)
= (!p + !q) * (q + p) (implication law)

But, here I get stuck... I've thought about applying DeMorgan's, but it seems like the two expressions won't be equivalent if I do this. I've verified this equivalence is true using truth tables, but the equivalence logic seems to be escaping me. Can anyone lend a hand?

Thanks! :)

-Max
 
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  • #2
A truth table would be much faster, but if you don't want that, you-ll have to develop (!p + !q) * (q + p), using the distributivity of *.
 

1. What is a biconditional equivalence?

A biconditional equivalence is a logical statement that connects two conditions with the words "if and only if." It means that both conditions must be true or false at the same time for the entire statement to be true.

2. How is a biconditional equivalence represented in symbolic logic?

A biconditional equivalence is represented by the symbol "↔" or "≡" in symbolic logic. The symbol is placed between the two conditions being connected.

3. What is the difference between a biconditional equivalence and a conditional statement?

A biconditional equivalence states that both conditions must be true or false at the same time, while a conditional statement only requires the first condition to be true for the entire statement to be true. A biconditional equivalence is a stronger statement than a conditional statement.

4. Can a biconditional equivalence be used to prove a statement?

Yes, a biconditional equivalence can be used to prove a statement by breaking it down into two conditional statements and proving each one separately. This is known as the "if and only if" proof method.

5. What are some common examples of biconditional equivalences?

Some common examples of biconditional equivalences include "A number is even if and only if it is divisible by 2," "A triangle is equilateral if and only if it has three equal sides," and "A person is a citizen of the United States if and only if they were born in the country or have gone through the naturalization process."

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