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.