The objects in a collection are dubbed the set"s **elements** or **members**. They room usually listed inside braces. We write $x \in A$ if $x$ is an facet (member) that a collection $A$.

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$\1,2,3\$ is a set with $3$ elements. That is the same as the set $\1,3,2\$ (order does no matter) and also the set $\1,1,2,3,3,3,3\$ (repetition does no matter). Typically, every objects room the exact same (for example, numbers), however they perform not need to be: $\ 1, 3, \textred, \textblue, \textJohn \$ is a set.

Ellipses are provided when a pattern is clear: $\1,2,3,4,\ldots,50\$ is the collection of all integers indigenous $1$ to $50$, inclusive.

Some sets we usage a lot:$\mathbbR$ is the collection of real numbers$\mathbbN$ is the collection of organic numbers$\mathbbZ$ is the set of integers$\mathbbQ$ is the collection of reasonable numbersIt is feasible to have actually a collection with no elements: $\\$. This is the **empty set** and is generally denoted $\emptyset$. This is **not** the exact same as $\ \emptyset \$, i beg your pardon is a collection with one facet (that wake up to be a (empty) set).

The variety of **distinct** aspects in a collection $S$ is called its **cardinality** and is denoted $|S|$. If $|S|$ is limitless (for example, $\mathbbZ$), us say the collection is **infinite**.

One common way to specify a collection is **set builder** notation. Below are 2 examples:

## Set Operations

Several operations can be performed on sets.**Union**: provided two set $A$ and also $B$, the

**union**$A \cup B$ is the set of all aspects that are in either $A$ or $B$. For example, if $A = \ 1,3,5 \$ and $B = \ 2,3,6 \$, climate $A \cup B = \ 1,2,3,4,5,6 \$. Note that $A \cup B = \\; (x \in A) \lor (x \in B) \$.

**Intersection**: given two sets $A$ and also $B$, the

**intersection**$A \cap B$ is the collection of all elements that room in both $A$ and also $B$. For example, if $A = \ 1,2,3,4 \$ and $B = \ 3,4,5,6 \$, then $A \cap B = \ 3,4 \$. Keep in mind that $A \cap B = \\; (x \in A) \land (x \in B) \$. We say that $A$ and also $B$ are

**disjoint**if $A \cap B = \emptyset$.

**Difference**: provided two set $A$ and also $B$, the

**difference**$A \setminus B$ is the set of all aspects that are in $A$ yet not in $B$. Because that example, if $A = \ 1,2,3,4 \$ and $B = \ 3,4 \$, climate $A \setminus B = \ 1,2 \$. Keep in mind that $A \setminus B = \ x \;$. $A \setminus B$ is also denoted $A - B$.

**Complement**: provided a set $A$, the

**complement**$\overlineA$ is the collection of all aspects that are

**not**in $A$. To define this, we require some an interpretation of the

**universe**the all feasible elements $U$. We can because of this view the complement as a special instance of collection difference, where $\overlineA = U \setminus A$. For example, if $U = \mathbbZ$ and $A = \\; x \text is one odd integer \$, climate $\overlineA = \\; x \text is an also number \$. Note that $\overlineA = \ x \;$.

**Cartesian Product**: given two set $A$ and $B$, the

**cartesian product**$A \times B$ is the collection of bespeak pairs whereby the first element is in $A$ and the 2nd element is in $B$. We have $A \times B = \\; a \in A \land b \in B \$. For example, if $A = \ 1,2,3 \$ and $B = \ a,b \$, climate $A \times B = \ (1,a),(1,b),(2,a),(2,b),(3,a),(3,b) \$.

## Subsets

A set $A$ is a **subset** that a set $B$ if every facet of $A$ is an element of $B$. We compose $A \subseteq B$. Another method of speak this is that $A \subseteq B$ if and also only if $\forall x\;(x \in A \rightarrow x \in B)$.

For any set S, we have:

$\emptyset \subseteq S$Proof: Must display that $\forall x\;(x \in \emptyset \rightarrow x \in S)$. Since $x \in \emptyset$ is constantly false, the implicit is always true. This is an example of a trivial or vacuous proof.

$S \subseteq S$Proof: Must present that $\forall x\;(x \in S \rightarrow x \in S)$. Deal with an element $x$. Us must show that $x \in S \rightarrow x \in S$. This implication is indistinguishable to $x \in S \lor x \notin S$, i m sorry is a tautology. Therefore, by global Generalization, $S \subseteq S$.

If $A \subseteq B$ and $A \neq B$, then us say $A$ is a**proper subset**of $B$ and also write $A \subset B$.

## Power Sets

The **power set** of a collection $A$ is the **set of all subsets** of $A$, denoted $\mathcalP(A)$. For example, if $A = \1,2,3\$, then$\mathcalP(A) = \\emptyset,\1\,\2\,\3\,\1,2\,\2,3\,\1,3\,\1,2,3\\$

Notice that $|\mathcalP(A)| = 2^$.

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## Set Equality

Two sets are equal if they contain the exact same elements. One means to present that 2 sets $A$ and also $B$ are equal is to display that $A \subseteq B$ and $B \subseteq A$:

\beginarraylllA \subseteq B \land B \subseteq A & \equiv & \forall x\;((x \in A \rightarrow x \in B) \land (x \in B \rightarrow x \in A)) \\&\equiv & \forall x\;(x \in A \leftrightarrow x \in B) \\&\equiv & A = B \\\endarrayNote: it is not enough to simply examine if the sets have the exact same size! lock must have actually **exactly** the very same elements. Remember, though, the order and also repetition carry out not matter.