LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

SUBSET OF FRACTION

__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 bu****t
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}. **