Properties of Set

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

If we change the order of the element in a set it will not change anything.
In other words the order in which the elements of a set are written is not important. Thus, the set {a1, b1, c1} can also be written as {a1, c1, b1} or {b1, c1, a1} or {b1, a1, c1} or {c1, a1, b1} or {c1, b1, a1}.

For Example:

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

i.e., { 6, 7, 8, 9} = {8, 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 ‘YODOO’ = {Y, O, D }

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,

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.

Reference : https://math.stackexchange.com/questions/2366063/prove-properties-of-sets-and-powersets