Question

Write down all the subsets of the set \[\left\{ {1,2,3} \right\}\].

Answer 1

Hint:
Here in this question, we will find the total number of subsets that can be obtained from a set of three elements using the formula \[{2^n}\]. Then, enlist all the subsets by considering the different combinations of elements, and keeping in mind the rules of set theory.

Complete step by step answer:
In this question, we are required to find the subsets of the given set. To do that, let us first find out the total number of subsets that can be formed with these three elements.
The total number of subsets is given by the formula, \[{2^n}\], where \[n\] is the total of elements in a given set.
Since we have 3 elements here, the total number of subsets is–
\[{2^n} = {2^3}\]
Now, we will apply the exponent on the term 2.
\[{2^n} = 8\]
Thus, there are a total of 8 subsets in our given set. Let us now start by enlisting these.
We will begin by enlisting all the single elements because each element of a set is also its subset.
\[\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\}\]
We will now enlist all the subsets with two terms.
\[\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\}\]
We will now enlist all the subsets with three terms. Since there are a total of three terms in our set, there will be one subset only. This is because every set is a subset of itself.
\[\left\{ {1,2,3} \right\}\]
We will now list \[\Phi \]. This is because the empty set \[\Phi \] is also a subset of every set.
\[\left\{ \Phi \right\}\]
Hence, all the subsets that can be formed from the set \[\left\{ {1,2,3} \right\}\] are –
\[\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\},\left\{ {1,2,3} \right\},\left\{ \Phi \right\}\].

Note:
A subset of a set is a collection of elements that are a part of another set. For example, if set A has \[\left\{ {a,b,c} \right\}\] and set B contains \[\left\{ {a,b} \right\}\] then set B will be the subset of set A. We might get confuse \[\left\{ {a,b} \right\}\] and \[\left\{ {b,a} \right\}\] as two different subsets of \[\left\{ {a,b,c} \right\}\]. However, \[\left\{ {a,b} \right\}\] or \[\left\{ {b,a} \right\}\] represent the same set only. The position of elements in a set does not matter, rather the elements themselves matters.