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 Y ⊆ X. so we can conclude that, X & Y are equal set and the equal sets X & Y are denoted by X = Y.