Question

If $A = \{ x:{x^2} - 5x + 6 = 0\} ,\,B = \{ 2,4\} ,\,C = \{ 4,5\} $ then $A \times (B \cap C)$ is
A. $\{ (2,4),(3,4)\} $
B. $\{ (4,2),(4,3)\} $
C. $\{ (2,4),(3,4),(4,4)\} $
D. $\{ (2,2),(3,3),(4,4),(5,5)\} $

Answer 1

Hint: If $A$ and $B$ are two non-empty sets, then the set of all ordered pairs of elements from $A$ and $B$ is known as their Cartesian product and is denoted by $A \times B$.
$A \times B = \{ (x,y):x \in A,y \in B\} $
In this question, we have to find the Cartesian product of the set $A$ and the set $B \cap C$. For that, we will find the elements in the set $A$ by finding the solution of the given quadratic equation and we will find the elements of $B \cap C$ by finding the intersection of the two sets.

Complete step by step solution:
We are given that $A = \{ x:{x^2} - 5x + 6 = 0\} ,\,B = \{ 2,4\} ,\,C = \{ 4,5\} $ and we have to find $A \times (B \cap C)$, so we will first find $B \cap C$ .
$B \cap C$ represents the intersection of the sets $B$ and $C$, that is, it is the set of elements that are present in both the sets $B$ and $C$.
We have $B = \{ 2,4\}$, that means the elements of this set are $2$ and $4$
And $C = \{ 4,5\}$, that means the elements of this set are $4$ and $5$ .
We see that the element $4$ is common in both the sets, so we get the intersection set as, $B \cap C = \{4\}$
Now, we are given $A = \{ x:{x^2} - 5x + 6 = 0\} $ , so to find the set $A$ , we solve the given equation.
We solve it as follows:
$
  {x^2} - 5x + 6 = 0 \\
  {x^2} - 2x - 3x + 6 = 0 \\
  x(x - 2) - 3(x - 2) = 0 \\
  (x - 3)(x - 2) = 0 \\
 $
Thus, we get:
$
  x = 2,3 \\
   \Rightarrow A = \{ 2,3\} \\
 $
Now, we find the product of the set $A$ and the set $(B \cap C)$.
$
  A \times (B \cap C) = \{ 2,3\} \times \{ 4\} \\
   \Rightarrow A \times (B \cap C) = \{ (2,4),(3,4)\} \\
 $
The correct option is option A.

Note: The intersection of two sets is a set that contains the common elements of both sets, while the union of two sets is a set that contains all the elements of both sets. Here, we have used the intersection of two sets, so one should not get confused between the two.