 UPSKILL MATH PLUS

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### Theory:

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 $\left\{1\right\}\in A$ then $\left\{1\right\}\in B$.

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

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, $n\left(A\right)\le n\left(B\right)$. 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 $A\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left\{1,2\right\}$.

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, $\varnothing \phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\left\{\right\}$ and $\left\{1,2\right\}$.

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

The singleton subsets are $\left\{1\right\}$ and $\left\{2\right\}$.

Therefore, the subsets of the set $$A$$ are $\left\{\right\},\left\{1\right\},\left\{2\right\}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\left\{1,2\right\}$.

2. Consider the set with three elements $B\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left\{a,b,c\right\}$.

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, $\varnothing \phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\left\{\right\}$ and $\left\{a,b,c\right\}$.

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

The singleton subsets are $\left\{a\right\}$, $\left\{b\right\}$ and $\left\{c\right\}$.

Let us write the subsets with two elements.

$\left\{a,b\right\},\phantom{\rule{0.147em}{0ex}}\left\{a,c\right\},\phantom{\rule{0.147em}{0ex}}\left\{b,c\right\}$.

Therefore, the subsets of the set $$B$$ are $\left\{\right\},\left\{a\right\},\phantom{\rule{0.147em}{0ex}}\left\{b\right\},\phantom{\rule{0.147em}{0ex}}\left\{c\right\},\left\{a,b\right\},\phantom{\rule{0.147em}{0ex}}\left\{a,c\right\},\phantom{\rule{0.147em}{0ex}}\left\{b,c\right\}$.