Set difference is one of the important operations on sets which can be used to find the difference between two sets.
Let us discuss this operation in detail.
Let X and Y be two sets.
Now, we can define the following new set.
X \ Y = {z | z ∈ X but z ∉ Y}
(That is z must be in X and must not be in Y)
X \ Y is read as "X difference Y"
Now that X \ Y contains only elements of X which are not in Y and the figure given below illustrates this.
Some authors use A - B for A \ B. We shall use the notation A \ B which is widely used in mathematics for set difference.
Question 1 :
For A = {5, 10, 15, 20} B = {6, 10, 12, 18, 24} and C = {7, 10, 12, 14, 21, 28} verify whether A\(B\C) = (A\B)\C. Justify your answer.
Solution :
A = {5, 10, 15, 20}
B = {6, 10, 12, 18, 24}
C = {7, 10, 12, 14, 21, 28}
L.H.S
A\(B\C)
(B\C) = {6, 10, 12, 18, 24}\{7, 10, 12, 14, 21, 28}
= {6, 18, 24}
A\(B\C) = {5, 10, 15, 20}\{6, 18, 24}
= {5, 10, 15, 20} ---(1)
(A\B) = {5, 10, 15, 20}\{6, 10, 12, 18, 24}
= {5, 15, 20}
(A\B)\C = {5, 15, 20} \ {7, 10, 12, 14, 21, 28}
= {5, 15, 20} --- (2)
A\(B\C) ≠ (A\B)\C
Question 2 :
Let A = {-5, -3, -2, -1} B = {-2, -1, 0} and C = {-6, -4, -2}. Find A\(B\C) and (A\B)\C. What can we conclude about set difference operation?
Solution :
A = {-5, -3, -2, -1} B = {-2, -1, 0} and C = {-6, -4, -2}
L.H.S
A\(B\C)
(B\C) = {-2, -1, 0}\{-6, -4, -2}
= {-1,0}
A\(B\C) = {-5, -3, -2, -1}\{-1, 0}
= {-5, -3, -2} ---- (1)
R.H.S
(A\B)\C
(A\B) = {-5, -3, -2, -1}\{-2, -1, 0}
= {-5, -3}
(A\B)\C = {-5,-3} \ {-6,-4,-2}
= {-5, -3} -----(2)
(1) ≠ (2)
A\(B\C) ≠ (A\B)\C
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