Subset 

Definition: 

A set X is said to be a subset of another set Y, If all elements of set X are elements of set Y. In other words, the set X is contained inside the set Y.

Symbol ‘⊂’ is denoted for ‘Subset of’ or is ‘Contained in’.

If set A is a subset of set B, then it is denoted as, A⊂B

If set A is a subset of set B, then it is denoted as, A⊆B

We write A ⊂ B instead of A ⊆ B.

If set B is a superset of set A, then it is denoted as, B ⊇A

A is subset of B, (A ⊂ B) in Venn diagram

Example 1: If, Set A = {a, b, c} and Set B = {a, c, b, d}. Is A a subset of set B. Why ?.

Solution: Set A = {a, b, c} and Set B = {a, b, c, d}

a is element of set A, and a is element of set B

b is also element of set A, and also of set B

c in set A, and also in set B

Elements {a, b, c} are present in set A and these are also present in set B.

We see that all the elements of set A, and every single one in B, so we can sat that A ⊂ B.

Element d is in set B, but d is not in set A. It doesn’t matter, we look only the elements in A.

So, A is a subset of set B.

∴ A ⊂ B

A set to be a subset of another set it needs to have all elements are present in the another set.

A is a subset of B, if and only if every element of A is in B.

Example 2: If A = {p, q, r} and B = {e, f, g, h, p, q, r}

Then, A is subset of B

(A ⊂ B) in Venn diagram

Example 3: Set A = {1, 2, 3, 4, 5, 6} and Set B = {1, 2, 3, 4}

Solution: Here, B = {1, 2, 3, 4},

We note that every element of B is also an element of A.
so, B is subset of A. So, we can say B ⊂ A

Example 4: If A = {l, m, n, o} and B = {s, t, u, v, l, m, n, o}

Then, A is subset of B, (A ⊂ B) in Venn diagram

Example 5: Let A is a set of all students in a school, and B is a set of all students in a class.

Solution: We note that every element of B is also an element of A, 
so, B is subset of A. B ⊂ A

Example 6: If A = {1, 2, 3}, B = {3, 4, 5}, C = {1, 2, 3, 4, 5} and D = {1, 5}. Is (i) A ⊂ C, (ii) A ⊂ B (iii) B ⊂ C (iv) A ⊂ D (v) D ⊂ B

Solution: A = {1, 2, 3}, C = {1, 2, 3, 4, 5}, Since, {1, 2, 3} are in set C

(i) ∴ A ⊂ C

A = {1, 2, 3}, B = {3, 4, 5}, Since, {1, 2, 3} are not in set B

(ii) ∴ A ⊄ B

B = {3, 4, 5}, C = {1, 2. 3, 4, 5}, Since, {3, 4, 5} are in set C

(iii) ∴ B ⊂ C

A = {1, 2, 3}, D = {1, 5}, Since, {1, 2, 3} are not in set D

(iv) ∴ A ⊄ D

D = {1, 5}, B = {3, 4, 5}, Since, {1, 5} are not in set B

(v) ∴ D ⊄ B

Example 7: If, Set A = {p, q, r} and Set B = {p, q, r, s}. Is A a subset of B. Why ?.

Solution: Set A = {p, q, r} and Set B = {p, q, r, s}

p is element of set A, and p is element of set B

q is also element of set A, and also of set B

r in set A, and also in set B

We see that all the elements of set A, and every single one in B, so we can sat that A ⊂ B.

Element s is in set B, but s is not is not in set A. It doesn’t matter, we look only the elements in A.

Elements {p, q, r} are present in set A and these are also present in set B.

So, A is a subset of set B.

∴ A ⊂ B

A set to be a subset of another set it needs to have all elements are present in the another set.

A is a subset of B, if and only if every element of A is in B.

Example 8: Set A = {1, 2, 3}, and Set B = {3, 6}. Is A is subset of B.

Solution: Subsets of A are {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3},{1, 2, 3}, { }.

But {3, 6} is not a subset of set A, because it has an element {6} which is not the element of set A

∴ A ⊄ B

If A is not subset of B, we write A ⊄ B

Proper Subset

Definition:

Set A is a proper subset of set B, if all the elements of set A are also the elements of set of B, but A is not equal to B i.e. at least one element of set B that is not an element of set A, (A ≠ B) then A is a proper subset of B and denoted by A ⊊ B or A ⊂ B.

B is a proper superset of A denoted by B⊋ A or B ⊃ A.

Note: When A is a proper subset of B then it is also a subset of B.

Example 9: If A = {1, 2, 3}, B = {1, 2}, C = {1, 2, 3} and D = {1, 5}. Find the relationship between set A and set B set C and set D.

Solution: Set B is a proper subset of A.

Set C is a subset of set A, but it is not a proper subset of A since A = C.

The set D is not even a subset of A, since 5 is not an element of A.

Example 10: If A = {a, b, c} and B = {a, b ,c, d}, Find the relationship between set A and set B.

Solution: Set A is equal to {a, b, c} is a subset of B, {a, b, c, d} but is not a proper subset of {a, b, c}.

∴ A ⊂ B

{a, b, c} is a proper subset of {a, b, c, d} because the element d is not in the first set.

Properties of empty set

1. Every set A is a subset of itself i.e.

A ⊂ A

2. Empty set ∅ has no elements, so that ∅ is a subset of every set.

Example 11: If X = {1, 3, 5} and Y = {1, 2, 4, 5, 6}. Find the relationship between set X and set Y.

Solution: Here we can say that X is not a subset of Y, since every element of X is not contained in Y.

We see that 3 is not an element of set Y (3 ∉ Y).

The statement ” X is not a subset of Y” is denoted by X ⊄ Y.

Number of subsets and Proper subsets of a set

Number of subsets in a set X is 2², where n is the number of elements in a set.

Example 12: If P = {x, y, z}, then write all possible subsets of P. Find the number of subsets.

Solution: The subset of P containing one element – {x} {y} {z}

The subset of P containing two elements – {x, y} {x, z} {y, z}

The subset of P containing three elements – {x, y, z}

The subset of P containing no element – { }

Therefore, number of all subsets of P is eight which is equal to 2³

Example 13: If A = {1, 2, 3}, then we write all the possible subsets of A and find their numbers.

Solution: The subset of A containing one elements each – {1},{2},{3}

The subset of A containing two elements each – {1, 2}, {1, 3}, {2, 3}

The subset of A containing three elements each – {1, 2, 3}

The subset of A containing no any elements each – { }

Therefore, number of all subsets of A is eight which is equal to 2³

Proper subsets are = {1}, {2}, {3}, {1, 2},{1, 3},{2, 3}, { }

Number of proper subsets are = 2³ – 1 = 8 – 1 = 7

Example 14: If the number of elements in a set is 3. Find the number of subsets and proper subsets.

Solution: Number of elements in a set = 3

Then the number of subsets = 3² = 9

and the number of proper subsets = 3² – 1 = 8

Example 15: If the number of elements in a set is 5. Find the number of subsets and proper subsets.

Solution: Number of elements in a set = 5

Then the number of subsets = 5² = 25

and the number of proper subsets = 5² – 1 = 24

Example 16: If X = {1, 3, 5, 7, 9} and Y = {3, 1, 7, 5, 9}. Find the relationship between set X and set Y.

Solution: Given, X = {1, 3, 5, 7, 9} and Y = {3, 1, 7, 5, 9}

The elements in set X and set Y are same only order in which they are appear is not same.

We know that order in which the elements appear in a set is not important.

∴ Set X and Set Y are equivalent.

Example 17: Which of the following is a subset of set A

Solution: A = {1, 3, 5, 7, 9},

X = {1, 7, 5}, Y = {1, 3, 9} and Z = {3, 1, 7, 5, 9}

All of above

Example 18: Which of the following is not a subset of set A.

Solution: A = {1, 2, 3, 4, 5},

X = {1, 7, 5}, Y = {1, 3, 4} and Z = {3, 1, 5, 9}

Set X and set Z are not the subset of set A.