Fill in the blanks:

Question

For all sets A and B, if A is in the set of B, then A union B in in the set of B.

Proof: Suppose A and B are any sets and A in in the set of B. [we must show that __________] Let x be and element of _______. [we must show that __________] by definition of union, x is an elment of ______ ______ x is an element of _______. In case x is an element of ________, then since A is in the set of B, x is an element of _________. In case x is an element of B, then clearly, x is an element of B. So in either case, x is an element of __________ [as was to be shown]

Answer 1

Proof: Suppose A and B are any sets and A in in the set of B. [we must show that A∪B ⊆B]

Let x be an element of A∪B. [we must show that x∈A or x∈B] By definition of union, x is an elment of ______ or x is an element of _______.

In case x is an element of ________, then since A is in the set of B, x is an element of _________.

In case x is an element of B, then clearly, x is an element of B.

So in either case, x is an element of __________ [as was to be shown]

Hope the above hints are enough for you to complete the proof.