Question

Write down all the subsets of the following sets $\{ 1,2,3,4,5,.....\infty \} $
1) $\{ a\} $
2) $\{ a,b\} $
3) $\{ 1,2,3\} $
4) $\emptyset $

Answer 1

Hint: If \[A\] and \[B\] are two sets, we define the subset “if \[A\] is a subset of \[B\] then, all members of \[A\] is a member of \[B\]”.Subset is denoted by \[A \subseteq B\] and \[\emptyset \] is the empty set.
Here we going to write subsets, for that, we use some rules,
The first rule is, “Empty set is a subset of every set”.
The second rule is, “The set itself is a subset of the set”.

Complete step-by-step answer:
1) \[\{ a\} \]
We can take \[A = \{ a\} \]
As we say the first rule, “empty set is a subset of every set”
That is
\[\emptyset \subseteq A\]
As we say the second rule, “the set itself is a subset for set”
\[\{ a\} \subseteq A\]
Hence \[\emptyset ,\{ a\} \] are the only subsets of \[A = \{ a\} \]

2) \[\{ a,b\} \]
We can take \[A = \{ a,b\} \]
As we say the first rule
\[\emptyset \subseteq A\]
The second rule,
\[\{ a,b\} \subseteq A\]
Also we can write,
\[\{ a\} \subseteq A\]
\[\{ b\} \subseteq A\]
Hence \[\emptyset ,\{ a,b\} ,\{ a\} ,\{ b\} \] are the subsets of set \[A = \{ a,b\} \]

3) \[\{ 1,2,3\} \]
We can take \[A = \{ 1,2,3\} \]
By the first rule,
\[\emptyset \subseteq A\]
In the second rule,
\[\{ 1,2,3\} \subseteq A\]
Then,
\[\{ 1\} \subseteq A\]
\[\{ 2\} \subseteq A\]
\[\{ 3\} \subseteq A\]
Hence \[\emptyset ,\{ 1,2,3\} ,\{ 1\} ,\{ 2\} ,\{ 3\} \] are the subsets of set \[A = \{ 1,2,3\} \]

4) \[\emptyset \]
It is different from other sets, \[\emptyset \] is the empty set.
Empty set has no elements inside the set.
Therefore it has no subsets.
Hence the set \[\emptyset \] has no subsets.

Note:The rule “Empty set is subset for every set” is called trivial subset.And the rule “the set itself is a subset of a set” is called improper subset.Here we can observe that the section (iv) has \[\emptyset \] is empty set it has no elements, But we can apply the first rule \[\emptyset \] is the subset for the set \[\emptyset \] (\[\emptyset \subseteq \emptyset \]) and also we apply the second rule \[\emptyset \] is the subset for the set \[\emptyset \] (\[\emptyset \subseteq \emptyset \]). Hence both rules are equal also it has no elements.