Question

If three sets are given as A=\left\\{ 1,2,3 \right\\},B=\left\\{ 3,4 \right\\},C=\left\\{ 4,5,6 \right\\}, find $A\times \left( B\cap C \right)$.

Answer 1

Hint: First of all we have to know about intersection and cartesian product of sets. We know that intersection of any two sets X and Y denoted $X\cap Y$, is the set containing all elements of X that also belong to Y. The cartesian product of sets x and y is denoted by $x\times y$ and it is the set of all possible ordered pairs in which the first element is of x and the second element is of y.
In set builder form x\times y=\left\\{ \left( a,b \right):a\in x\ and\ b\in B \right\\}.

Complete step-by-step answer:
We have been given A=\left\\{ 1,2,3 \right\\},B=\left\\{ 3,4 \right\\},C=\left\\{ 4,5,6 \right\\} and asked to find the value of $A\times \left( B\cap C \right)$.
We know that intersection of any two sets x and y is the set containing all elements of x that also belong to y.
So, we have B=\left\\{ 3,4 \right\\}\ and\ C=\left\\{ 4,5,6 \right\\}.
Since, the only element common to both sets is 4, we have,
B\cap C=\left\\{ 4 \right\\}
Also, we know that the cartesian product of any two sets x and y is denoted by $x\times y$. It is the set whose elements are all the possible ordered pairs in which the first element is of x and the second element is of y.
Now, we have A=\left\\{ 1,2,3 \right\\}\ and\ \left( B\cap C \right)=\left\\{ 4 \right\\}.
\begin{aligned} & \Rightarrow A\times \left( B\cap C \right)=\left\\{ 1,2,3 \right\\}\times \left\\{ 4 \right\\} \\\ & =\left\\{ \left( 1,4 \right),\left( 2,4 \right),\left( 3,4 \right) \right\\} \\\ \end{aligned}

Note: Remember that the cartesian product of any two sets x and y, is the set of all possible ordered pairs in which the first element is of x and the second element is of y. If by mistake, a student writes $\left( y,x \right)$ instead of $\left( x,y \right)$ as the ordered pairs, then the answer will go wrong. Also, the intersection of any two sets x and y is the set that contains all elements of x that also belong to y. Students should not get confused with union and intersection concepts.