Equal Sets

Equal sets are sets in set theory in which the number of elements is the same and all elements are equal.

Definition

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

Examples:

(i) Let $A=\{1,2,3,4\}$ and $B=\{4,3,2,1\}$. Then $A=B$.

(ii) Prove that $A=\{x:x$ is prime such that $1<x<10\}$ and $B=\{2,3,5,7\}$are equal sets.

Solution: $A=\{x:x$ is prime such that $1<x<10\}=\{2,3,5,7\}$

Now, the number of elements in $A$ and $B$ are the same, i.e., $4$ and all the elements are also equal.

Therefore, $A=B$

$A=\{x:x$ is prime such that $1<x<10\}=\{2,3,5,7\}=B$