# Properties of Sets

What are the two basic properties of sets?

The two basic properties to represent a set are explained below using various examples.

1. The change in order of writing the elements does not make any changes in the set.

In other words the order in which the elements of a set are written is not important. Thus, the set {a, b, c} can also be written as {a, c, b} or {b, c, a} or {b, a, c} or {c, a, b} or {c, b, a}.

For Example:

Set A = {4, 6, 7, 8, 9} is same as set A = {8, 4, 9, 7, 6}

i.e., {4, 6, 7, 8, 9} = {8, 4, 9, 7, 6}

Similarly, {w, x, y, z} = {x, z, w, y} = {z, w, x, y}    and so on.

2. If one or many elements of a set are repeated, the set remains the same.

In other words the elements of a set should be distinct. So, if any element of a set is repeated number of times in the set, we consider it as a single element. Thus, {1, 1, 2, 2, 3, 3, 4, 4, 4} = {1, 2, 3, 4}

The set of letters in the word ‘GOOGLE’ = {G, O, L, E}

For Example:

The set A = {5, 6, 7, 6, 8, 5, 9} is same as set A= {5, 6, 7, 8, 9}

i.e., {5, 6, 7, 6, 8, 5, 9} = {5, 6, 7, 8, 9}

In general, the elements of a set are not repeated. Thus,

(i) if T is a set of letters of the word ‘moon’: then T = {m, o, n},

There are two o’s in the word ‘moon’ but it is written in the set only once.

(ii) if U = {letters of the word ‘COMMITTEE’}; then U = {C, O, M, T, E}

Solved examples using the properties of sets:

1. Write the set of vowels used in the word ‘UNIVERSITY’.

Solution:

Set V = {U, I, E}

2. For each statement, given below, state whether it is true or false along with the explanations.

(i) {9, 9, 9, 9, 9, ……..} = {9}

(ii) {p, q, r, s, t} = {t, s, r, q, p}

Solution:

(i) {9, 9, 9, 9, 9, ……..} = {9}

True, since repetition of elements does not change the set.

(ii) {p, q, r, s, t} = {t, s, r, q, p}

True, since the change in order of writing the elements does not change the set.

Set Theory

Sets

Objects Form a Set

Elements of a Set

Representation of a Set

Different Notations in Sets

Standard Sets of Numbers

Types of Sets

Pairs of Sets

Subset

Subsets of a Given Set

Operations on Sets

Union of Sets

Intersection of Sets

Difference of two Sets

Complement of a Set

Cardinal number of a set

Cardinal Properties of Sets

Venn Diagrams