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 numbers

It 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:

$\mathbbR = \ r \;$$O = \ x \;$

## 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 \\\endarray

Note: 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.