Show that f is a bijection where f : A×B -> B×A be defined as f ((a, b)) =(b, a) where A and B are two non empty sets 

Asked by Rituparna | 15th Jun, 2017, 11:59: AM

Expert Answer:

begin mathsize 16px style The space function space is space one minus one
Let comma space open parentheses straight a subscript 1 comma straight b subscript 1 close parentheses comma space open parentheses straight a subscript 2 comma straight b subscript 2 close parentheses element of straight A cross times straight B space be space such space that space straight f open parentheses open parentheses straight a subscript 1 comma straight b subscript 1 close parentheses close parentheses equals straight f open parentheses open parentheses straight a subscript 2 comma straight b subscript 2 close parentheses close parentheses
rightwards double arrow open parentheses straight b subscript 1 comma straight a subscript 1 close parentheses equals open parentheses straight b subscript 2 comma straight a subscript 2 close parentheses rightwards double arrow straight b subscript 1 equals straight b subscript 2 space and space straight a subscript 1 equals straight a subscript 2
rightwards double arrow open parentheses straight a subscript 1 comma straight b subscript 1 close parentheses equals open parentheses straight a subscript 2 comma straight b subscript 2 close parentheses
Thus comma space straight f open parentheses open parentheses straight a subscript 1 comma straight b subscript 1 close parentheses close parentheses equals straight f open parentheses open parentheses straight a subscript 2 comma straight b subscript 2 close parentheses close parentheses rightwards double arrow open parentheses straight a subscript 1 comma straight b subscript 1 close parentheses equals open parentheses straight a subscript 2 comma straight b subscript 2 close parentheses rightwards double arrow straight f space is space one minus one
The space function space straight f space is space onto
Consider space any space element space open parentheses straight b comma straight a close parentheses element of straight B cross times straight A comma space then space straight b element of straight B comma space straight a element of straight A
rightwards double arrow open parentheses straight a comma straight b close parentheses element of straight A cross times straight B
straight f space is space onto.
Hence comma space given space function space is space bijective. end style

Answered by Sneha shidid | 15th Jun, 2017, 01:16: PM