Subset | Universal set

Subset | Universal set

Let's suppose, A and B are two non empty sets. The set A is said to be the subset of B if every element of A is contained in the set B. It is denoted by A ⊆ B.

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For example:

Let, A={a,b,c} and B={a,b,c,d,e}. 

Then, A is subset of B. A ⊆ B and read as A is subset of B or A is contained in B. “If A is a subset of B then B is a super set of A”.

Proper Subset

Let X be a subset of Y. Then X is said to be the proper subset of Y if X contains at least one element less than Y.

For example:

Let, X={1,2,3} and Y={1,2,3,4,5}. Then, X is a proper subset of Y. It is written as X ⊂ Y.

Improper subset

Let X be subset of Y. Then X is said to be the improper subset of Y if X and Y are equal sets. i.e. X and Y contain same elements or every element of X is contained in Y.

For example:

Let, X = {a, e, i, o, u} and Y = {e, a, i, o, u}. Then, X is an improper subset of Y. It is denoted by X ⊆ Y.

Note:

i.          The number of subsets of a given set = 2ⁿ where n is the number of elements contained in the set.

ii.         The number of a proper subset of a given set = 2ⁿ – 1 where n is the number of elements contained in a set.

iii.        An empty or null set is a subset of every set.

Universal Set

A set which contains all the sets under consideration as the subsets is called a universal set. It is denoted by the capital letter U of the English alphabet.

For example:

Let, A = {a set of girl students}

       B = {a set of boy students}

       C = {a set of class 8 students}

       D = {a set of class 9 students}

Then the universal set of all these sets is U = {a set of all students of the school}.

Workout Examples

Example 1: Find all the subsets of the following sets.

a.     A = {a}

b.     B = {a, b}

c.     C = {1, 2, 3}

Solution: a. Here,

                           A = {a}

                           Subsets are: { }, {a}

                b. Here,

                           B = {a, b}

                           Subsets are: { }, {a}, {b}, {a, b}

                c. Here,

                           C = {1, 2, 3}

                           Subsets are: { }, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}

               

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