00:01
Okay, so in this question we have r is symmetric if and if r is equal to our inverse so what we want to do is we want to prove both forwards and backwards direction because this is an if and only if statement so first we'll prove the forward direction so we first assume that r is b t r i c symmetric so by definition of our inverse we have this is equal to b a on the condition that a b is in r.
00:35
But because r is symmetric, this is equivalent to saying if a, b is in r, then this implies that b a is in r.
00:48
So we can just swap this statement out to this statement.
00:53
So this is equivalent to writing b, a, on the condition, b, a is in r.
01:00
Now we just swap the labels around, so we swap b to a and a to b.
01:05
So then this is just a, b on the condition that a, b is in r, and this is quite literally a definition of r.
01:15
So therefore, they are equal to itself.
01:19
So now we want to prove that r is equal to r inverse...