State whether the following are equal or equivalent sets :
${1, 2, 3}; {+, -, \times}$

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Solution

Step 1: Checking whether two sets are equal
Let,
$A = {1, 2, 3} B = {+, -, \times}$
Equal sets:
Two sets $A$ and $B$ are said to be equal if they have exactly the same elements and we write $A = B$ .
And a set does not change if one or more elements of the sets are repeated.
All equal sets are equivalent sets.
$A = {1, 2, 3}$ and $B = {+, -, \times}$ are not equal since each element of $A$ is not in $B$ and vice-versa.
$A \neq B (∵ 1 \in A but 1 \notin B)$
Hence, the given sets are not equal.
Step 2: Checking whether two sets are equivalent
Equivalent sets:
Two sets $A$ and $B$ are equivalent to each other if the number of elements in both sets is equal i.e $n (A) = n (B)$ .
And if two sets are equivalent we write $A ~ B$ or $A \equiv B$
All equivalent sets are not equal sets.
Here,
$\begin{array}{rcl} A & = & {1, 2, 3} \end{array}$
$\begin{array}{rcl} ∴ n (A) & = & 3 \end{array}$
and,
$\begin{array}{rcl} B & = & {+, -, \times} \end{array}$
$∴ n (B) = 3$
$∵ n (A) = n (B)$ , So, the set $A$ is equivalent to set $B$ , i.e $A ~ B$
Hence, the given sets are equivalent.

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