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Let A and B be the two sets such that n(A−B)=60+3x, n(B−A)=8x and n(A∩B)=x−4 then draw a Venn diagram to illustrate this information. If n(A)=n(B) then find
(a) The value of x
(b) n(A∪B)

Answer
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Hint: We solve this problem by using the Venn diagrams of sets. The Venn diagrams represent the diagrammatic representation of sets inside the universal set ′μ′
For solving the first part we use the given condition n(A)=n(B) along with the formulas of sets that is
n(A)=n(A−B)+n(A∩B)n(B)=n(B−A)+n(A∩B)
For solving second part we use the general formula of sets that is
n(A∪B)=n(A)+n(B)−n(A∩B)

Complete step-by-step solution
We are given that n(A−B)=60+3x, n(B−A)=8x and n(A∩B)=x−4
Let us draw a Venn diagram that represents the given information then we get
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(a) The value of x
We are given that
⇒n(A)=n(B).......equation(i)
We know that the formulas of sets that is
n(A)=n(A−B)+n(A∩B)n(B)=n(B−A)+n(A∩B)
By using the above formulas to equation (i) we get
⇒n(A−B)+n(A∩B)=n(B−A)+n(A∩B)⇒n(A−B)=n(B−A)
By substituting the required values in above equation we get
⇒60+3x=8x⇒5x=60⇒x=12
Therefore, the value of x is 12
(b) n(A∪B)
We know that the direct formula of union of sets that is
n(A∪B)=n(A)+n(B)−n(A∩B)
By substituting the required values from the formulas we used before in above equation we get
⇒n(A∪B)=(n(A−B)+n(A∩B))+(n(B−A)+n(A∩B))−n(A∩B)⇒n(A∪B)=n(A−B)+n(B−A)+n(A∩B)
Now by substituting the required values in terms of x in above equation we get
⇒n(A∪B)=60+3x+8x+x−4⇒n(A∪B)=12x+56
Now, by substituting x=12 in above equation we get
⇒n(A∪B)=12×12+56⇒n(A∪B)=200
Therefore the value of n(A∪B) is 200.

Note: Students may make mistakes in the Venn diagram representation.
Venn diagrams are the diagrammatic representation of sets in the universal set ′μ′
So the Venn diagram must be drawn as
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But students may miss the universal set ′μ′ and draw the Venn diagram as
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This will be the wrong representation because all the sets are subsets of a universal set ′μ′ which is very important to represent in the Venn diagram.