Question

Show that:
\[\neg(p \leftrightarrow q) \equiv p \leftrightarrow \neg q\]

Answer 1

truth tables is the easiest way

Answer 2

indeed. but how to prove without truth tables?

Answer 3

oh ok.

Answer 4

im thinking...
\[\neg(\neg p \leftrightarrow q)\]
\[p \leftrightarrow \neg q\]
but im not sure if that's right

Answer 5

i'm not sure if \[p \leftrightarrow \equiv \neg p \leftrightarrow q\]
which i doubt is true

Answer 6

\[ \large \neg(p\leftrightarrow q)\equiv\neg[(p\to q)\wedge(q\to p)] \]
\[ \large \equiv[\neg(\neg p\vee q )\vee\neg(\neg q\vee p)] \]
\[ \large \equiv[(p\wedge\neg q)\vee(q\wedge\neg p)] \]
\[ \large \equiv[(p\vee q)\wedge(p\vee\neg p)]\wedge
[(\neg q\vee q)\wedge(\neg q\vee\neg p)] \]
\[ \large \equiv[(p\vee q)\wedge\mathbb{V}]\wedge
[\mathbb{V}\wedge(\neg q\vee\neg p)] \]
\[ \large \equiv(p\vee q)\wedge(\neg q\vee\neg p) \]
\[ \large \equiv(\neg q\to p)\wedge(p\to\neg q) \]
\[ \large \equiv p\leftrightarrow\neg q \]

Answer 7

wait..is that a definition? \[p \leftrightarrow \equiv (p \rightarrow q) \wedge (q \rightarrow p)\]

Answer 8

u could say so

Answer 9

ahh incredible. wouldn't have thought of this