1.5 Equal Sets

Equal sets: Given two sets A and B, if every element of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal. The two equal sets will have exactly the same elements.

Two sets A and B are said to be equal if they have exactly the same elements and we write \(\mathrm{A}=\mathrm{B}\). Otherwise, the sets are said to be unequal and we write \(\mathrm{A} \neq \mathrm{B}\).

Let us consider the following examples:
(i) Let \(\mathrm{A}=\{1,2,3,4\}\) and \(\quad \mathrm{B}=\{3,1,4,2\}\). Then \(\mathrm{A}=\mathrm{B}\).

Note: A set does not change if one or more elements of the set are repeated. For example, the sets \(\mathrm{A}=\{1,2,3\}\) and \(\mathrm{B}=\{2,2,1,3,3\}\) are equal, since each element of \(\mathrm{A}\) is in \(\mathrm{B}\) and vice-versa (the order in which elements in a set are represented does not matter). That is why we generally do not repeat any element in describing a set.

Example 7: Find the pairs of equal sets, if any, give reasons:
\(
\begin{array}{ll}
\mathrm{A}=\{0\}, & \mathrm{B}=\{x: x>15 \text { and } x<5\} \\
\mathrm{C}=\{x: x-5=0\}, & \mathrm{D}=\left\{x: x^{2}=25\right\}
\end{array}
\)
\(\mathrm{E}=\left\{x: x\right.\) is an integral positive root of the equation \(\left.x^{2}-2 x-15=0\right\}\).

Solution: Since \(0 \in A\) and 0 does not belong to any of the sets \(B, C, D\) and \(E\), it follows that, \(\mathrm{A} \neq \mathrm{B}, \mathrm{A} \neq \mathrm{C}, \mathrm{A} \neq \mathrm{D}, \mathrm{A} \neq \mathrm{E}\).

Since \(\mathrm{B}=\phi\) but none of the other sets are empty. Therefore \(\mathrm{B} \neq \mathrm{C}, \mathrm{B} \neq \mathrm{D}\) and \(\mathrm{B} \neq \mathrm{E}\). Also \(\mathrm{C}=\{5\}\) but \(-5 \in \mathrm{D}\), hence \(\mathrm{C} \neq \mathrm{D}\).

Since \(\mathrm{E}=\{5\}, \mathrm{C}=\mathrm{E}\). Further, \(\mathrm{D}=\{-5,5\}\) and \(\mathrm{E}=\{5\}\), we find that, \(\mathrm{D} \neq \mathrm{E}\). Thus, the only pair of equal sets is \(\mathrm{C}\) and \(\mathrm{E}\).