00:01
So the given term is we are considering a function from a to b.
00:06
We have to prove that.
00:08
We have to prove that f is injective.
00:13
F is injective.
00:19
If and only if, x is equal to f inverse of f of x.
00:24
F inverse of f of x for all x contained in e for all x contained in e.
00:33
So first let us assume that, let us assume that x is equal to f inverse of f of x, f of x for all x contained in e, for all x contained in e.
00:53
So now let small x comma y belongs to capital x x comma y belongs to capital x such that f of x is equal to f of y f of x is equal to f of y in order to show that f is injective f is injective we have to show that x is equal to y x is equal to y so let us consider capital x is equal to singleton x then taking functions on both sides that is f of x is equal to f of x this also in terms of singleton now since we know that since we know that f of x is equal to f of y we can write f of capital x is equal to singleton f of instead of f of x we are writing f of y now which implies this can be rewritten as y is equal to f inverse of f of x if inverse of x now by our assumption that x is equal to f inverse of f of x where y belongs to x is equal to singleton x is equal to singleton x which implies y is equal to x.
02:32
Y is equal to x.
02:34
This shows that f is injective.
02:37
Conversely, let us assume that f is injective already.
02:43
Now let us assume that x belongs to f inverse of f of x.
02:49
F of f of x.
02:52
So which implies f of x is equal to f of x belongs to f of capital x...