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 = 22
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
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