00:01
In this problem we are given that f is a function from set x to set y, we are asked to prove that f is one to one.
00:10
If and only if, for any subset a subset of x, f inverse of f of a is equal to a.
00:20
Now, to prove this, first consider the following definitions.
00:24
For any a subset of x, we have f of a is the set of all y element of y sets that.
00:31
Exist an x element of a with the property that f of x is equal to y similarly for any z subset of y we have f inverse of z is the set of all x element of a such that f of x is it now first assume that f is one to one then by property of one to one functions we have for any x x2 element of x with the property that f of x1 is equal to f of x2 then this implies that x1 is equal to x2 now we will prove that for any a subset of x inverse of f of a is equal to a for that first consider x element of f inverse of f of a then by definition this means that there exists a element of f of a with the property that f of x is equal to y.
01:45
Now by definition if y belongs to the set f of a then that there exists an inside element of a with the property that y is equal to f of z.
02:06
Now from these two we get f of x is equal to f of z.
02:13
But since f is 1 -1, this implies that x is equal to is said.
02:20
And since z is an element of a, from here we get x is also an element of a...