CLASS-8SUBSET & SUPERSET

Subset & Superset

If A & B are two sets such that every elephant of A is an elephant of B, we say that A is a subset of B or A is included in B. we express this in symbols as A B, If the set A is not a Subset of the Set B, we write A B

Let, A = { 2, 3 }, and B = { 2, 3, 4, 5 }, then  2 A and 2 B, also 3 A and 3 B, hence every element of A is an element of B. so, A ⊆  B.

However then 4 A and 4 B, also 5 A, and 5 B, so every element of B do not belong to or not an element of A. So, B is not a subset of A, which is to mentioned like B A.

Please reminder that, each set is a subset of itself, thus for any set A, A ⊆ A or for any set B, B ⊆ B.

It has been considered that, the null set φ is or will be a subset of every set.

In the examples we have considered,

1) The number of subsets of the set A = 4 = 2² = 2ⁿ  ( where n number of elements of A )

2) The number of subsets of the set B =  8 = 2ᶟ = 2ⁿ  ( where n number of elements of B )

3) The number of subsets of the set C =  16 = 2⁴ = 2ⁿ  ( where n number of elements of B )

If the number of elements of a finite set X is n, the number of the subsets of X = 2ⁿ

Proper Subsets -

If the sets X & Y are such that, where every element of X is supposed to be an element of Y but Y has at least one element which is not an element of X then X is called a proper subset of Y, this is expressed in symbols as X Y.

Super Set –

Let X be a subset of Y, i.e, X Y, then we say that B is a superset of A. We express this in symbols as Y X

Example.-1) let X = { 3, 5, 7 } and Y = { 1, 2, 3, 4, 5, 6, 7, 8 }, then Y X

Equal Set –

Two sets X & Y are suppose to be equal if each element of X is an element of Y and each element of Y is an element of X, in other words, The Set X & Y are equal whenever X is a subset of Y and Y is a subset of X, that is  X  Y and   X. so we can conclude that, X & Y are equal set and the equal sets X & Y are denoted by X = Y.