Show that if A is an infinite set, then whenever B is a set. A ? B is also an infinite set.

Solution:Step1To findWe have to show that if A is an infinite set, then whenever B is a set. A B is also an infinite set.Step2The union of sets A and B can be written asWe have to take all possible cases of set B.Case1:-Suppose A be an infinite set and B= then =AHence, A is an infinite set.Case2:-Assume B=A then is an infinite set.Case3:-Assume B is a finite set thenIs also an infinite set because A is infinite.So, will always infinite.Case4:-Assume if B is an infinite set thenIs also an infinite set as A and B are infinite sets.So, will be also infinite.Hence, from above all the cases it is clear that is an infinite set.