Subsets
If two sets X & Y, where every member of ‘X’ is also a member of ‘Y’ then we can say that, ‘X’ is a subset of ‘Y’ then this is to be written like X ⊆ Y, the set ‘X’ is a subset of ‘Y’ can also be expressed by saying ‘Y’ is a superset of ‘X’, we can write like Y ⊇ X.
Example- 1
Let, X = { 3, 5, 6, 7 }, Y = { 3, 4, 5, 6, 7 }, Z = { 1, 2, 4, 8, 9 }
3 ϵ X & 3 ϵ Y ; 5 ϵ X & 5 ϵ Y ; 6 ϵ X & 6 ϵ Y ; 7 ϵ X & 7 ϵ Y ; whereas 4 ∉ X but 4 ϵ Y,
So, we can see that X ⊆ Y and also, Y ⊇ X
And, 3 ϵ X but 3 ∉ Z, 5 ϵ X but 5 ∉ Z, 6 ϵ X but 6 ∉ Z, 7 ϵ X but 7 ∉ Z, whereas
1 ∉ X but 1 ϵ Z ; 2 ∉ X but 2 ϵ Z ; 4 ∉ X but 4 ϵ Z ; 8 ∉ X but 8 ϵ Z ; 9 ∉ X but 9 ϵ Z
So, we can see that X ⊄ Z
Also, 3 ϵ Y but 3 ∉ Z ; 4 ϵ Y but 4 ∉ Z ; 5 ϵ Y but 5 ∉ Z ; 6 ϵ Y but 6 ∉ Z ; 7 ϵ Y but 7 ∉ Z, where as
1 ∉ Y but 1 ϵ Z ; 2 ∉ Y but 2 ϵ Z ; 4 ∉ Y but 4 ϵ Z ; 8 ∉ Y but 8 ϵ Z ; 9 ∉ Y but 9 ϵ Z ;
So, we can see that Y ⊄ Z
Example – 2
Let, A = { y / y is a letter of the word BISWADEEP }
And B = { y / y is a letter of the word BISWA } , then prove that B ⊆ A.
Ans.) A = { B, I, S, W, A, D, E, P } and B = { B, I, S, W, A }
You can see from above that, members of Set ‘B’ belongs to members of Set ‘A’
So, it’s proven that B ⊆ A.
Example – 3
Write all the subset of the sets of the set { 0, 2, 4, 5, 6 }
Ans.) There are five numbers in { 0, 2, 4, 5, 6 }, so subset of one members each will be {0}, {2}, {4}, {5}, {6}.
The subsets of two members each will be {0,2}, {0,4}, {0,5}, {0,6}, {2,4}, {2,5}, {2,6}, {4,5}, {4,6}, {5,6}.
Also, it has been considered that, the empty set φ and the set itself are two subsets of the set, the required subset are φ, {0}, {2}, {4}, {5}, {6}, {0,2}, {0,4}, {0,5}, {0,6}, {2,4}, {2,5}, {2,6}, {4,5}, {4,6}, {5,6}.