Equal Sets

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≠B.

[Note] A set does not change if one or more elements of the set are repeated. For example, the sets C={1,2,3} and X={2,2,1,3,3} are equal, since each element of C is in X and vice—versa. That is why we generally do not repeat any element in describing a set.

Definition: Equal Sets
Set A is equal to set B. symbolized by A=B. if and only if set A and set B contain exactly the same elements.

For example, if set D={1,2,3} and set E={3,1,2}, then D=E because they contain exactly the same elements. The order of the elements in the set is not important. If two sets are equal, both must contain the same number of elements. The number of elements in a set is called its cardinal number.
🌈 Set Cardinality — the Number of Elements of a Set
⛲ Example 1. From the sets given below, select equal sets:
F={2,4,8,12}, G={1,2,3,4}, H={4,8,12,14}, J={3,1,4,2}, K={-1,1}, L={0,a}, M={1,-1}, N={0,1}
✍ Solution:
It can be seen that

8∈F, 8∉G, 8∉J, 8∉K, 8∉L, 8∉M, 8∉N
⇒ F≠G, F≠J, F≠K, F≠L, F≠M, F≠N

Also, 2∈F, 2∉H
∴ F≠H

3∈G, 3∉H, 3∉K, 3∉L, 3∉M, 3∉N
∴ G≠H, G≠K, G≠L, G≠M, G≠N

12∈H, 12∉J, 12∉K, 12∉L, 12∉M, 1∉N
∴ H≠J, H≠K, H≠L, H≠M, H≠N

4∈J, 4∉K, 4∉L, 4∉M, 4∉N
∴ J≠K, J≠L, J≠M, J≠N

Similarly, K≠L, K≠M, K≠N, L≠M, L≠N, M≠N.
The order in which the elements of a set are listed is not significant.
∴ G=J and K=M.
Hence, among the given sets, G=J and K=M.

⛲ Ex2. From the following sets, select equal sets.
O={2,4,6,8}, P={1,2,3,4,5}, Q={-2,4,6,8}, R={2,3,5,4,1}, S={8,6,2,4}.
🌟 Two sets are said to be equal, if they have exactly the same elements.
✍ Solution:
O={2,4,6,8} and S={8,6,2,4}
Since, each element of set O presents in set S and vice—versa.
Therefore, O and S are equal sets.
Similarly, P={1,2,3,4,5} and R={2,3,5,4,1}
Since, each element of set P presents in set R and vice—versa.
Therefore, P and R are also equal sets.

⛲ Ex3. Find the pairs of equal sets, if any, give reasons:
T={0}, V={x:x>15 and x<5}, W={x:x-5=0}, Y={x:x²=25},
Z={x:x is an integral positive root of the equation x²-2x-15=0}.
✍ Solution:
Since 0∈T and 0 does not belong to any of the sets V, W, Y and Z, it follows that, T≠V, T≠W, T≠Y, T≠Z.
Since V=Ø but none of the other sets are empty. Therefore V≠W, V≠Y and V≠Z. Also W={5} but -5∈Y, hence W≠Y.

Since Z={5}, W=Z. Further, Y={-5,5} and Z={5}, we find that, Y≠Z. Thus, the only pair of equal sets is W and Z.

🌈 Equal sets are a type of sets, so, find other types through this link. Sets and Their Types — Types of Sets