Let ′A′ denote the set with a cup
′B′ denote the set with a plate
′A ∩ B′ denote the set with both a cup and a plate
′A ∪ B′ denote the set with the trays either a cup or a plate
n( X) denote the number of elements in the set ′X′
Given, total number of trays n(A ∪ B) = 25Number of trays that contain cups n(A) = 15
Number of trays that contain cups n(B) = 21
To find the trays with both a cup and a plate n(A ∩ B),
We know that
n(A ∪ B) = n(A)+ n(B) − n(A ∩ B)
Rearranging the terms, we get
n(A ∩ B) = n(A)+ n(B) − n(A ∪ B)
From the above,
n(A ∩ B) = 15 + 21 − 25
= 11
n(A ∩ B) = 11
Therefore, number of trays with both a cup and a plate is ′11′.