Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

α|==|β iff |= (α↔β)

Does anyone know how to go about showing that this is true?

share|improve this question
    
Truth table?${}$ –  peterwhy Sep 14 '13 at 18:17

1 Answer 1

up vote 2 down vote accepted

HINT:

First, assume that $\alpha\leftrightarrow\beta$ is a tautology. Suppose that $g$ is some assignment of truth values to the relevant propositional variables. Since $\alpha\leftrightarrow\beta$ is true in this assignment, conclude that if $\alpha$ was true then $\beta$ is true, and vice versa.

In the other direction, assume that $\alpha$ is logically equivalent to $\beta$. Take some assignment $g$ and show that if $\alpha$ is false then $\beta$ is false, and so $\alpha\leftrightarrow\beta$ is true; and if $\alpha$ is true then $\beta$ is true, and so $\alpha\leftrightarrow\beta$ is true. These are the only two options, so $\alpha\leftrightarrow\beta$ is a tautology, as wanted.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.