92 lines
3.0 KiB
Markdown
92 lines
3.0 KiB
Markdown
A set is a collection of _unordered_ elements.
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A set cannot contain duplicates.
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## Notation
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### Set Notation
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Declaration of a set $A$ with elements $a$, $b$, $c$:
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$$ A := {a, b, c} $$
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### Cardinality
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Amount of Elements in a set $A$
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Notation: $|A|$
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$$
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A := {1, 2, 3, 4} \
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|A| = 4
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$$
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### Well-Known Sets
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- Empty Set: $emptyset = {}$
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- Natural Numbers: $N = {1, 2, 3, ...}$
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- Integers: $ZZ = {-2, -1, 0, 1, 2}$
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- Rational Numbers: $QQ = {1/2, 22/7 }$
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- Real Numbers: $RR = {1, pi, sqrt(2)}$
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- Complex Numbers: $CC = {i, pi, 1, sqrt(-1)}$
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### Set-Builder Notation
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Common form of notation to create sets without explicitly specifying elements.
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$$
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A := {x in N | 0 <= x <= 5} \
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A = {1, 2, 3, 4, 5}
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$$
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### Member of
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Denote whether $x$ is an element of the set $A$
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Notation: $x in A$
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Negation: $x in.not A$
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### Subsets
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| Type | Explanation | Notation |
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| ------------------------ | ---------------------------------------------------------------------- | ------------------- |
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| **Subset** | Every element of $A$ is in $B$ | $A subset B$ |
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| **Subset or equal to** | Every element of $A$ is in $B$, or they are the exactly same set | $A subset.eq B$ |
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| **Proper subset** | Every element of $A$ is in $B$, but $A$ is definitely smaller than $B$ | $A subset.sq B$<br> |
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| **Superset**<br> | $A$ contains everything that is in $B$ | $A supset B$ |
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| **Superset or equal to** | $A$ contains everything that is in $B$, or they are identical | $A supset.eq B$ |
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## Operations
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### Union
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Notation: $A union B$
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Definition: all elements from both sets _without adding duplicates_
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$$ A := {1, 2, 3}\ B := {3, 4, 5}\ A union B = {1, 2, 3, 4, 5} $$
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### Intersection
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Notation:$A inter B$
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Definition: all elements _contained in both sets_
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$$
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A := {1, 2, 3} \
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B := {2, 3, 4} \
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A inter B = {2, 3}
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$$
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### Difference
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Notation: $A backslash B$
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Definition: all elements _in $A$ that are not in $B$_
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$$
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A := {1, 2, 3} \
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B := {3, 4, 5} \
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A backslash B = {1, 2}
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$$
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### Symmetric Difference
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Notation: $A Delta B$
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Definition: all elements _only in $A$ or only in $B$_
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$$
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A := {1, 2, 3} \
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B := {2, 3, 4} \
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A Delta B = {1, 4}
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$$
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### Cartesian Product
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Notation: $A times B$
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Definition: all pairs of all elements in $A$ and $B$
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$$
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A := {1, 2} \
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B := {3, 4, 5} \
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A times B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
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$$
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### Powerset
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Notation: $cal(P)(A)$
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Definition: all possible _Subsets of A_
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$$
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A := {1, 2, 3} \
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cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
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$$
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## Closures
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A **closed set** under the binary operation $*$ denoted by $forall a,b in S, a * b in S$
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"All $a * b$ from set $S$ are included in set $S$"
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A set can be closed under any binary operation ($+, -, *, \/$), depending on the set itself if it actually is closed or not. $NN$ for example is not closed under subtraction.
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