Chapter 1: Sets

Exercise

Question

Question: 6 (i)

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Show that for any sets A and B,

(i) A = (A ∩ B) ∪ (A - B)

Answer

1
Divide the elements of A into two groups: those that belong to B and those that do not belong to B.
Venn diagram showing the standard decomposition of two sets into three disjoint regions: A − B, A $\cap$ B, and B − A.
Set A is divided into two parts: A ∩ B and A − B.
Elements of A that also belong to B: A $∩$ B
Elements of A that do not belong to B: A − B
2
Combine the two groups obtained in Step 1 to reconstruct set A.
Answer: Every element of A belongs to either A $∩$ B or A − B. Therefore, A = (A $∩$ B) $∪$ (A - B).
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