Subsets — lesson. Mathematics State Board, Class 9.

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Let $A$ and $B$ are two sets. If every element of $A$ is also an element of $B$, then $A$ is a subset of $B$. It is denoted as

A \subseteq B

We can read

A \subseteq B

as '$A$ is a subset of $B$'.

Suppose

A \subseteq B

and

{1} \in A

then

{1} \in B

If $A$ is not a subset of $B$, then we can write

A ⊄ B

Important!

If $A$ is a subset of $B$, the number of elements in the set $A$ must be less than or equal to the number of elements in the set $B$. That is, $n (A) \leq n (B)$ . Since every element of $A$ is also an element of $B$, the set $B$ must have at least as many elements as $A$, thus $n(A) ≤ n(B)$. It can be concluded that if $A$ is a subset of $B$, then the cardinal number of $A$ must be less than the cardinal number of $B$.
If $A \subseteq B$ and $B \subseteq A$ , then $A = B$.
An empty set is a subset of every set.
Every set is a subset of itself.

Example:

1. Consider the set with two elements

A = {1, 2}

By the concept, 'Empty set is a subset of every set' and 'Every set is a subset of itself' the two obvious subsets of the set are the empty set and the whole set itself.

That is,

\emptyset or {}

and

{1, 2}

Now let us write the elements and its combination of subsets.

The singleton subsets are

{1}

and

{2}

Therefore, the subsets of the set $A$ are

{}, {1}, {2} and {1, 2}

2. Consider the set with three elements

B = {a, b, c}

By the concept, 'Empty set is a subset of every set' and 'Every set is a subset of itself' the two obvious subsets of the set are the empty set and the whole set itself.

That is,