Sets - Subsets

A subset is a set of elements that are also in another set. Recall that a set is a collection of distinct elements. For example, $\{\text{cat}, \text{dog}, \text{fish}, \text{bird}\}$ is a set containing a few animals, $\{2,4,6,8,10\}$ is a set of even numbers, and $\{a, b, c, d\}$ is a set of letters.

For a given set $B$, the set $A$ is a subset of $B$ if every element that is in $A$ is also in $B$. This is denoted by $A \subseteq B$.

Here is a set $A$ that contains all of the integers in the range 0 to 10: $A = \{0,1,2,3,4,5,6,7,8,9,10\}$. Create a subset of $A$, called $B$, such that $B$ contains all of the odd numbers of $A$.

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Select all of the odd numbers in $A$ and add them to $B$:

$$B = \{1,3,5,7,9\}.$$

$B$ is a subset of $A$ because all of the elements that are in $B$ are also in $A$. $_\square$

$A = B$ if and only if $A \subseteq B$ and $B \subseteq A$.

$A$ is a proper subset of $B$ if $A$ is a subset of $B$ and $A$ is not equal to $B$. This is denoted by $A \subset B$. The empty set $\emptyset$ is a proper subset of every non-empty set.

Subset versus proper subset:

Take the following sets: $A = \{a,b,c,d,e\}$, $B = \{a,b,c,d,e\}$, and $C = \{a,d,e\}$.

• $A \subseteq B$ because every element in $A$ is also in $B$.
• $B \subseteq A$ because every element in $B$ is also in $A$.
• $C \subseteq A$ and $C \subseteq B$ because every element in $C$ is also in $A$ and $B$.
• $C \subset A$ and $C \subset B$ because all elements in $C$ are in $A$ and $B$, but $C$ is not equal to $A$ or $B$.

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

What are all of the subsets of $\{ 1, 2, 3 \}$?

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The subsets are $\emptyset$ (the empty set), $\{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\},\{1,2,3\}$. There are 8 of them. $_\square$

Note: You can use the rule of product to show that for a finite set $A$, there are $2^{ |A| }$ subsets.

If set $A$ is the set containing the first 10 prime numbers, how many subsets does $A$ have?

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Since set $A$ contains 10 distinct elements, we see that $| A | = 10$. Using the rule of product, we see that the number of subsets of $A$ are $2^{|A|} = 2^{10} = 1024$. Thus, there are 1024 subsets of set $A$. $_\square$

• Supersets

• Power sets