Equivalence of propositions $a\Leftrightarrow b$ and $(a\Rightarrow b)\wedge(\neg b\Rightarrow \neg a)$.

Question

Is the statement $a\Leftrightarrow b$ equivalent to the statement $(a\Rightarrow b)\wedge(\neg b\Rightarrow \neg a)$?

For example: "I get out if and only if it is sunny" should be equivalent to "When I get out, it's sunny. When it's not sunny, I don't get out".

Edit. Sorry, I just realized what I have is simply $(b\Rightarrow a)\wedge(\neg b\Rightarrow \neg a)$ which should be good now (simply as you guys have answered $(\neg b\Rightarrow \neg a)$ is equivalent by contraposition to $a\Rightarrow b$).

No. You've written down $a \Rightarrow b$ twice. You need to get $b \Rightarrow a$ involved. — Randall, Oct 16 '20 at 15:06
$a\implies b$ is logically equivalent to $\neg b\implies \neg a$. This is known as contraposition. Your statement on the right then was no different than just $a\implies b$. — JMoravitz, Oct 16 '20 at 15:08

score 2 · Accepted Answer · answered Oct 16 '20 at 15:10

2

No, its not. $a\Rightarrow b$ and $\neg b\Rightarrow \neg a$ are equivalent. One is the contraposition of the other. What you mean is that $a\Rightarrow b\wedge b\Rightarrow a$.

answered Oct 16 '20 at 15:10

Wuestenfux

20,964

score 1 · Answer 2 · answered Oct 16 '20 at 15:09

1

$(\neg b\implies \neg a)$ is equivalent to the statement $(a\implies b)$.

$a\iff b$ is equivalent to the statement $(a\implies b)\wedge(b\implies a )$ which is equivalent to $(a\implies b)\wedge(\neg a\implies \neg b )$.

answered Oct 16 '20 at 15:09

QED

12,644

Equivalence of propositions $a\Leftrightarrow b$ and $(a\Rightarrow b)\wedge(\neg b\Rightarrow \neg a)$.

2 Answers2