Q3 of 86 Page 26

Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.

It is given in the question that,


And,


We have to show that, B = C


Let us now assume,


So,



∴


Now, when then:



∴


As,


So,


∴



∴


Similarly, it can be shown that C ⊂ B


Hence, B = C


More from this chapter

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1

Decide, among the following sets, which sets are subsets of one and another:

A = {x : x ∈ R and x satisfy x2 – 8x + 12 = 0},


B = {2, 4, 6}, C = {2, 4, 6, 8, . . .}, D = {6}

 2

In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example:

(i) If x ∈ A and A ∈ B, then x ∈ B


(ii) If A ⊂ B and B ∈ C, then A ∈ C


(iii) If A ⊂ B and B ⊂ C, then A ⊂ C


(iv) If A ⊄ B and B ⊄ C, then A ⊄ C


(v) If x ∈ A and A ⊄ B, then x ∈ B


(vi) If A ⊂ B and x ∉ B, then x ∉ A

 4

Show that the following four conditions are equivalent:

(i) A ⊂ B (ii) A – B = ϕ (iii) A ∪ B = B


(iv) A ∩ B = A

 5

Show that if A ⊂ B, then C – B ⊂ C – A.