ICSE Class 7 Maths Chapter 35 Subsets Universal Set

Chapter 35: Subsets, Universal Set (Including Operations On Sets)

35.1 Subset

If each element of set A is also present in set B, then set A is said to be the subset of set B. Conversely, if B is subset of A, then each element of set B is present in set A.

For example:

(i) A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6, 7} implies that A is a subset of B, as each element of set A is in set B also.

(ii) A = {6, 7, 8, 9, 10} and B = {7, 9, 10} implies B is subset of A and so on.

Symbolically, if A is a subset of B, we write: \(A \subseteq B\).

and, if B is a subset of A, we write: \(B \subseteq A\).

35.2 Super-set

When set A is the subset of set B, then set B is called the super-set of A.

Symbolically, we write: \(B \supseteq A\).

\(1. A \subseteq B\) is read as: "A is subset of B" or "A is contained in B".

\(2. B \supseteq A\) is read as: "B is super-set of A" or "B contains A".

\(3. A \not\subseteq B\) means "A is not a subset of B". And \(B \not\supseteq A\) means "B is not a super-set of A".

\(4.\) Every set is a subset of itself, i.e., \(A \subseteq A\), \(B \subseteq B\) and so on.

\(5.\) Empty set is subset of every set, i.e., \(\phi \subseteq A\), \(\phi \subseteq B\) and so on.

Example 1:

Find subsets of: (i) \(\{ \}\) (ii) \(\{a, b\}\)

Solution:

Since, every set is subset of itself and the empty set is subset of every set:

(i) Subset of \(\{ \} = \{ \}\)

(ii) Subsets of \(\{a, b\} = \{ \}, \{a\}, \{b\}\) and \(\{a, b\}\)

35.3 Proper Subset

All the subsets, other than the set itself, are called proper subsets. The symbol for proper subset is \(\subset\), i.e., if A is proper subset of B, we write: \(A \subset B\).

\(1.\) No set is proper subset of itself.

\(2.\) When set A is proper subset of set B:

(i) each element of set A is in set B.

(ii) number of elements in set A is less than the number of elements in set B.

Example 2:

Find, if possible, all proper subsets of: (i) \(\{ \}\) (ii) \(\{a\}\) (iii) \(\{a, b\}\).

Solution:

Since, no set is proper subset of itself, therefore:

(i) \(\{ \}\) has no proper subset.

(ii) Proper subset of \(\{a\} = \{ \}\)

(iii) Proper subsets of \(\{a, b\} = \{ \}, \{a\}\) and \(\{b\}\).

35.4 Number Of Subsets And Number Of Proper Subsets Of A Given Set

If a set has \(n\) elements in it,

(i) the number of its subsets = \(2^n\) and

(ii) the number of its proper subsets = \(2^n - 1\) [No set is proper subset of itself]

e.g., Number of elements in \(\{a, b\} = 2\) \(\Rightarrow n = 2\)

No. of its subsets = \(2^n = 2^2 = 4\)

and no. of its proper subsets = \(2^n - 1 = 2^2 - 1 = 3\)

Teacher's Note

When organizing items in a closet, some shirts are part of a larger collection - just like subsets relate to larger sets in mathematics.

35.5 Universal Set

Any set, which contains all the elements of various sets under discussion, is called the universal set.

For example:

If the sets under discussion are: A = {2, 3, 5}, B = {5, 6, 9, 12} and C = {3, 6, 9, 12}, then form a set which contains every element of sets A, B and C.

Clearly, the set obtained is {2, 3, 5, 6, 9, 12}. So this set is called the universal set for the sets A, B and C under discussion.

A universal set is represented by the symbol \(\xi\) (read pxi) or U.

Thus, \(\xi = \{2, 3, 5, 6, 9, 12\}\)

Note: Universal set for the sets under consideration is not unique, i.e., we may have more than one universal set for the same sets under consideration.

Thus, for the sets A, B and C, given above,

(i) {1, 2, 3, 4, ..., 15} can be taken as universal set, since it contains every element of the sets under discussion.

(ii) Set N, the set of natural numbers, can also be taken as universal set, since each element of the sets under discussion is a natural number and so on.

\(1.\) Every set under discussion is a subset of the universal set.

\(2. \xi \subseteq \xi\), because every set is a subset of itself.

Teacher's Note

A universal set is like the entire school - specific groups like the math club are subsets within it.

35.6 Complement Of A Set

The complement of a set A is the set of elements, which are present in the universal set but are not present in set A.

The complement of a set A is written as A', and is read as complement of set A.

Thus, if universal set, \(\xi = \{1, 2, 3, 4, 5, 6, 7, 8\}\),

A = {1, 3, 5, 7} and B = {2, 4, 6};

then: (i) Complement of set A = A' = {elements which are in \(\xi\) but not in A}

= {2, 4, 6, 8}

and, (ii) Complement of set B = B' = {1, 3, 5, 7, 8}

\(1.\) The set and its complement are always disjoint, i.e., sets A and A' are disjoint, sets B and B' are disjoint sets and so on.

\(2.\) The complement of the empty set is the universal set, i.e., \(\phi' = \xi\).

\(3.\) The complement of universal set is the empty set, i.e., \(\xi' = \phi\).

Teacher's Note

A complement is like identifying who is NOT on the guest list given a complete list of all potential guests.

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