Question

Let U the universal set be the set of all students of a school. A is a set of boys. B is a set of girls, and C is the set of students participating in sports. Describe the following set in words and represent it by Venn diagram
$\text{A}\bigcup \left( \text{B}\bigcap \text{C} \right)$

Answer 1

Hint: Union of the two sets A and B is a set which contains all the elements of A and B and none of the elements which are not present in both the sets. Hence if x is in Union of A and B then x is in A or x is in B.The intersection of two sets A and B is a set which contains all the elements present in both A and B and none of the elements which are not present in either of the two sets A and B. Hence if x is in the intersection of A and B then x is in both A and B.

Complete step-by-step answer:
Keeping in mind, the above definitions of union and intersection think what elements does the set $\text{A}\bigcup \left( \text{B}\bigcap \text{C} \right)$ contain.
Draw the corresponding Venn diagram. Note that the set of girls and Boys are disjoint

Let x is in $\text{A}\bigcup \left( \text{B}\bigcap \text{C} \right)$.
Hence from the definition of the union of two sets we have x is in A or x is in $\text{B}\bigcap \text{C}$.
Hence from the definition of the intersection of two sets, we have x is in A or (x is in B and x is in C).
Hence either x is a boy or x is a girl playing sports.
Hence $\text{A}\bigcup \left( \text{B}\bigcap \text{C} \right)$ is the set of all boys and sports playing girls.
The Venn diagram is as shown below.
The shaded areas represent the set $\text{A}\bigcup \left( \text{B}\bigcap \text{C} \right)$

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Note: The construction of the Venn diagram was sequential.
We first visualise what A looks like and what $\text{B}\bigcap \text{C}$ looks like.

The shaded area is the set A.

The shaded area is $\text{B}\bigcap \text{C}$.
Hence the Venn diagram of $\text{A}\bigcup \left( \text{B}\bigcap \text{C} \right)$ is