Question

Are the following pairs of sets equal? Give reasons.
(i). $A=\{2,3\},B=\{x:x\text{ is a solution of }{{\text{x}}^{2}}+5x+6=0\}$ .
 $\begin{align}
(ii). & A=\{x:x\text{ is a letter of the word ''WOLF''}\} \\
 & B=\{x:x\text{ is the letter of the word ''FOLLOW''}\} \\
\end{align}$

Answer 1

Hint: In order to solve this question, we need to first convert the given sets to their roaster form. Then you need to compare the pair of sets to check whether they are equal or not. Remember for two sets to be equal the number of elements is equal and all the elements are the same.

Complete step-by-step answer:

Before starting with the solution, let us discuss different symbols and operations related to sets.
Union: The union (denoted by $\cup $ ) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
Intersection: The intersection of two sets has only the elements common to both sets. If an element is in just one set it is not part of the intersection. The symbol is an upside down $\cap $ .
Two sets are said to be equal if they have the all there elements equal.

Now let us start with the solution of the question. First, to solve part (i), we let us convert set B to roster form. It is given that $B=\{x:x\text{ is a solution of }{{\text{x}}^{2}}+5x+6=0\}$ . So, the possible values of x are the roots of the equation ${{\text{x}}^{2}}+5x+6=0$ .
$\begin{align}
  & \therefore {{\text{x}}^{2}}+5x+6=0 \\
 & \Rightarrow {{x}^{2}}+2x+3x+6=0 \\
 & \Rightarrow \left( x+2 \right)\left( x+3 \right)=0 \\
\end{align}$
Therefore, we can say that set $B=\left\{ -2,-3 \right\}$ , which is not equal to set $A=\{2,3\}$ .
Now let us move to part (ii). Let us start by converting set A to roster form. It is given that $A=\{x:x\text{ is a letter of the word ''WOLF''}\}$ . So, the possible values of x are W, O, L, F. Therefore, we can say that set $A=\left\{ W,O,L,F \right\}$ .
Now we will convert set B to its roaster form. It is given that $B=\{x:x\text{ is the letter of the word ''FOLLOW''}\}$ . So, the possible values of x are F, O, L, W. Therefore, we can say that set $B=\left\{ F,O,L,W \right\}$ . Hence, for part (ii), set A is equal to set B.

Note: We have used the fact that when the two sets are said to be equal, and also the definition of the given terms are also useful. One must memorize the definition so that there can be no mistake in the future. This is a simple question, and so the chance of making silly mistakes in a hurry to solve it are also higher. Also, be careful that the order of terms of the sets doesn’t matter for two sets to be equal. For example: set A={1,2} and set B={2,1} are equal sets. You must also know that equal sets and equivalent sets are two different things. For two sets to be equivalent, the number of terms in each set must be equal, but there is no condition that the elements must be the same.