2.1: Subsets and Equality

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For this section, let A and B be sets.

Subsets

Definition: Subset

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: Equal Set

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

Proper Subsets

Definition: Subset

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

Example \(\PageIndex{2}\):

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

\(B \subset A\). since \(3 \notin B, B \ne A.\)
\(C \subseteq A\), and \(C = A\).
\(D \nsubseteq A\) 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 \(0 \notin \mathbb{N}\), \(\mathbb{N} \nsubseteq \mathbb{W}\).

Power Sets

Definition: Power set

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

Example \(\PageIndex{4}\):

Let set \(A = \{Alex, Billy, Casey\}\)

\(P(A)\):

{Alex}	{Alex, Billy}	{Alex, Billy, Casey}
{Billy}	{Alex, Casey}	\(\emptyset\)
{Casey}	{Billy, Casey}

Cardinality

Definition: Cardinality

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 infinite set.

Example \(\PageIndex{5}\):

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

Example \(\PageIndex{6}\):

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

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 \(B \subseteq A\)