How can I prove that $a\implies b$ equals $\neg b\implies\neg a$ with truth tables?

Question

How can I prove that $a\implies b$ equals $\neg b\implies\neg a$ with truth tables?

I've made a separate truth table for each but I am unsure on how to make it into a single/combined truth table..

EDIT: would this then be the correct way to prove it? Suggestion 1:

Suggestion 2:

Answer 1

You can't compare two statements unless you put them in the same truth table. So you should put your tables together in one big table of six columns and four rows so that the $a$ and $\lnot a$ columns correspond correctly, and same for $b$ and $\lnot b$, then compare the $a\to b$ and $\lnot b\to\lnot a$ columns.

Answer 2

For all values of a,b you have shown the two expressions have the same value. Thus the two are logically equivalent. If you do not understand why, then make a truth table for
(a implies b) iff (not b implies not a).

It will tell you the same thing, that the two expressions have the same value for all choices of a,b.