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# 2.1: Subsets and Equality

• • Contributed by Pamini Thangarajah
• Professor (Mathematics & Computing) at Mount Royal University
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For this section, let A and B be sets.

## Subsets

Definition

A is a subset of B, (denoted $$A \subseteq B$$), if every element of A is also an element of B.

TEMPLATE To prove $$A \subseteq B$$:

We NEED to show If $$x \in A$$ then $$x \in B.$$

Example $$\PageIndex{1}$$:

Let $$A$$ be a set. Then $$\emptyset$$ and $$A$$ are subsets of $$A$$.

So $$A, \, \emptyset \subseteq A$$.

We can use set notation to specify and help describe our standard number systems. The following first four standard sets are given from smallest to biggest:

• $$\mathbb{N}$$ represents the set of all natural numbers: $$\mathbb{N} = \{1, 2, 3, 4...\}$$
• $$\mathbb{W}$$ represents the set of all whole numbers: $$\mathbb{W} = \{0, 1, 2, 3...\}$$
• $$\mathbb{Z}$$ represents the set of all integers: $$\mathbb{Z} = \{ ...-2, -1, 0, 1, 2...\}$$. $$i$$ is not used because it is used for complex numbers.
• $$\mathbb{Q}$$ represents the set of all rational numbers: $$\mathbb{Q} = \{0, \pm1, \pm\frac{1}{2}, \pm\frac{1}{3}...\}$$
• $$\mathbb{Q}$$c represents the set of all irrational numbers
• $$\mathbb{R}$$ represents the set of all real numbers
• $$\mathbb{U}$$ represents the universal set, the set to which all others are a subset.

## Equal Sets

Definition

A is equal to B, denoted $$A = B$$, if $$A \subseteq B$$ and $$B \subseteq A$$.

## Proper Subsets

Definition

A is a proper subset of B (denoted $$A \subset B$$) if $$A \subseteq B$$ and $$A \neq B$$.

Example $$\PageIndex{2}$$:

Let $$A = \{1, 3, 5\}, \, B = \{1, 5\}, \, C = \{1, 3, 5\}, \, D = \{1, 4\}$$

1. $$B \subset A$$. since $$3 \notin B, B \ne A.$$
2. $$C \subseteq A$$, and $$C = A$$.
3. $$D \nsubseteq A$$ because $$4 \notin A$$.

Example $$\PageIndex{3}$$:

Consider $$\mathbb{N}$$ and $$\mathbb{W}$$, the sets of natural and whole numbers.

$$\mathbb{N} \subset \mathbb{W}$$ because all elements of $$\mathbb{N}$$ are present in $$\mathbb{W}$$.

However, since $$0 \notin \mathbb{N}$$, $$\mathbb{N} \nsubseteq \mathbb{W}$$.

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## Power Sets

Definition

Let $$A$$ be a set. Then the set of all subsets of $$A$$ is called power set of $$A$$, and is denoted by $$P(A)$$.​​​​​​​

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Example $$\PageIndex{4}$$:

Let set $$A = \{Alex, Billy, Casey\}$$

$$P(A)$$:

 {Alex} {Alex, Billy} {Alex, Billy, Casey} {Billy} {Alex, Casey} $$\emptyset$$ {Casey} {Billy, Casey}
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## Cardinality

Definition

Let $$A$$ be a set. then the number of elements in the set $$A$$ is called cardinality of the set $$A$$, and is denoted by $$|A|$$ or $$n(A)$$. If $$n(A)$$ is finite then $$A$$ is called finite set, otherwise, it is called infini​​​​​​​te set.​​​​​​​

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Example $$\PageIndex{5}$$:

Let $$A = \{1, 2, 3, 4, 5, 6, 7 \}$$. Then $$\mid A \mid = 7$$.

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Example $$\PageIndex{6}$$:

Let $$A$$ be a set with $$|A|=n$$. Then $$|P(A)|=2^n.$$

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New Notations & Definitions

$$\subseteq$$: denotes that a set is a subset of another set.

$$\subset$$: denotes that a set is a proper subset of another set.

$$\mid$$: denotes for "such that" or "divides," depending on context.

{ } or $$\emptyset$$: denotes an empty set

Equal sets: $$A = B$$ if $$A \subseteq B$$ and $$B \subseteq A$$

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