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Saturday, October 9, 2021

2021/082) If,for all sets A, A \cup B = A then prove that B = \emptyset

This is true.

To prove the same we have A \cup B = A iff B \subseteq A

Let us take 2 sets A_1,A_2 which are disjoint and because it is true for every set A_1 \cup B = A_1 so B \subseteq A_1

and A_2 \cup B = A_2 so B \subseteq A_2

So from above 2 we have

B \subseteq A_1 \cap A_2

Because A_1,A_2 are disjoint sets so we have A_1 \cap A_2= \emptyset

So B = \emptyset

 

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