Definition of Subset:
If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B and we write it as A ⊆ B or B ⊇ A
The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’
• Every set is a subset of itself, i.e., A ⊂ A, B ⊂ B.
• Empty set is a subset of every set.
• Symbol ‘⊆’ is used to denote ‘is a subset of’ or ‘is contained in’.
• A ⊆ B means A is a subset of B or A is contained in B.
• B ⊆ A means B contains A.
For example;
1. Let A = {2, 4, 6}
B = {6, 4, 8, 2}
Here A is a subset of B
Since, all the elements of set A are contained in set B.
But B is not the subset of A
Since, all the elements of set B are not contained in set A.
Notes:
If ACB and BCA, then A = B, i.e., they are equal sets.
Every set is a subset of itself.
Null set or ∅ is a subset of every set.
2. The set N of natural numbers is a subset of the set Z of integers and we write N ⊂ Z.
3. Let A = {2, 4, 6}
B = {x : x is an even natural number less than 8}
Here A ⊂ B and B ⊂ A.
Hence, we can say A = B
4. Let A = {1, 2, 3, 4}
B = {4, 5, 6, 7}
Here A ⊄ B and also B ⊄ C
[⊄ denotes ‘not a subset of’]
Super Set:
Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B ⊇ A.
Symbol ⊇ is used to denote ‘is a super set of’
For example;
A = {a, e, i, o, u}
B = {a, b, c, ............., z}
Here A ⊆ B i.e., A is a subset of B but B ⊇ A i.e., B is a super set of A
Proper Subset:
If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B. The symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B.
For example;
1. A = {1, 2, 3, 4}
Here n(A) = 4
B = {1, 2, 3, 4, 5}
Here n(B) = 5
We observe that, all the elements of A are present in B but the element ‘5’ of B is not present in A.
So, we say that A is a proper subset of B.
Symbolically, we write it as A ⊂ B
Notes:
No set is a proper subset of itself.
Null set or ∅ is a proper subset of every set.
2. A = {p, q, r}
B = {p, q, r, s, t}
Here A is a proper subset of B as all the elements of set A are in set B and also A ≠ B.
Notes:
No set is a proper subset of itself.
Empty set is a proper subset of every set.
Power Set:
The collection of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.
For example;
If A = {p, q} then all the subsets of A will be
P(A) = {∅, {p}, {q}, {p, q}}
Number of elements of P(A) = n[P(A)] = 4 = 2 × 2
In general, n[P(A)] = 2m where m is the number of elements in set A.
Universal Set
A set which contains all the elements of other given sets is called a universal set. The symbol for denoting a universal set is ∪ or ξ.
For example;
1. If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7}
then U = {1, 2, 3, 4, 5, 7}
[Here A ⊆ U, B ⊆ U, C ⊆ U and U ⊇ A, U ⊇ B, U ⊇ C]
2. If P is a set of all whole numbers and Q is a set of all negative numbers then the universal set is a set of all integers.
3. If A = {a, b, c} B = {d, e} C = {f, g, h, i}
then U = {a, b, c, d, e, f, g, h, i} can be taken as universal set.
● Set Theory
● Sets
7th Grade Math Problems
From Subset to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
May 19, 24 03:36 PM
May 19, 24 03:19 PM
May 19, 24 02:23 PM
May 19, 24 01:26 PM
May 19, 24 10:42 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.