Set Theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects
Basic Operations
Subset
A set is a subset of another set if all its elements are also member of the other set: \(A \subseteq B = \forall a \in A (a \in B)\).
A proper, or strict subset adds the additional requirement that the sets may not be equal: \(A \subset B = A \subseteq B \land A \neq B\).
Intersection
An intersection represents the elements that are members of both sets: \(A \cap B = \{ x \mid x \in A \land x \in B \}\).
 \(A\) and \(B\) are disjoint if \(A \cap B = \emptyset\)
Union
A union represents the elements that are members of at least one of the sets: \(A \cup B = \{ x \mid x \in A \lor x \in B \}\).
Difference
A difference represents the elements that are members of one set, but not of the other: \(A \setminus B = \{ x \mid x \in A \land x \notin B \}\).
Symmetric Difference
A symmetric difference represents the elements that are member of any of the sets, but not of both: \(A \triangle B = (A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B)\).
Cardinality
The number of elements in the set \(A\) is expressed as \(\vert A \vert\). For example, \(\vert\{ 1, 2, 3 \}\vert = 3\). There also exists the alternate notation \(\#A = \vert A \vert\).
Complement
Given a set \(P \subseteq X\), the complement of \(P\) is the set of all elements of \(X\) not in \(P\). This is expressed as \(\overline{P} = \{ x \in X \mid x \notin P \}\). Basically, if \(P \subseteq X\), \(P \cup \overline{P} = X\). Also note that \(\overline{P} = X \setminus P\). As an example, if \(X = \{ 1, 2, 3 \}\) and \(P = \{ 1, 3 \}\), then \(\overline{P} = \{ 2 \}\).
Empty Set
 No element is member of the empty set: \(\forall x (x \notin \emptyset) \)
 The empty set is a subset of every set: \(\forall A (\emptyset \subseteq A)\)
 Any universal quantification over the empty set is true: given proposition \(P\), \(\forall x \in \emptyset \{P\}\) is true
Power Set
A power set is the set of all possible subsets of a set. Its defined like this: \(\wp (A) = \{ x \mid x \subseteq A \}\). Also denoted as \(\mathbb{P}A\). Note that the empty set is a member of all power sets: \(\forall A (\emptyset \in \wp (A))\). Thus, the power set of the empty set is the set of the empty set: \(\wp (\emptyset) = \{ \emptyset \}\).
If a set \(A\) has \(n\) elements, then \(\wp(A)\) has \(2^{n}\) elements.
Properties
 The power set operation is distributive over \(\cap\) but not over \(\cup\). \(\wp (A \cap B) = \wp (A) \cap \wp (B)\) but \(\wp (A \cup B) \neq \wp (A) \cup \wp (B)\)
Finite Power Set
A set containing all the finite subsets of a set \(X\) is denoted as \(\mathbb{F} X\). If \(X\) is a finite set, then \(\mathbb{F}X = \mathbb{P}X\).
Multi Set
The multiset (or bag) is the generalization of the concept of a set, where multiple ocurrences of an element are permitted. For example, \([ A, A, B ]\) and \([ A, B ]\) (notice the square brackets) are equal sets, but are considered different multisets.
The multiplicity of an element is the number of times an element occurs in a multiset. In the case of \([ A, A, B ]\), \(A\) has a multiplicity of 2, while \(B\) has a multiplicity of 1.
The multiset cardinality consists of the number of elements in the multiset, including duplicate ocurrences.
Notice that a set is considered a multiset, and for any nonempty set there are infinite multisets with different number of ocurrences for each of its elements.
A multiset can be defined as a relation that maps the elements in the bag to positive integers denoting multiplicity.
For example, a multiset of elements \(A\), \(B\), and \(C\), where multiplicity is 3, 1, and 2, respectively, can be denoted as \(\{ (A, 3), (B, 1), (C, 2) \}\). Elements that are not in the multiset are left out, rather than mapped to a multiplicity of zero.
Augmented Set
Given set \(X\), the augment set of \(X\) is \(X^{\perp}\), which is the union of the given set and the undefined element: \(X \cup \perp\).
Set Families
Set families are sets of other sets. They may be indexed, such that \(F = \{ A_{i} \mid i \in I \}\), where each \(A_{i}\) is a set.
Operations
 The expression \(\cap F\) represents the intersection of all the sets in the family \(F\):
It can be defined as follows: \(\cap F = \{ x \mid \forall A \in F (x \in A) \} = \cap_{i \in I} A_{i}\). Note that this operation is undefined if \(F = \emptyset\).
 The expression \(\cup F\) represents the union of all the sets in the family \(F\):
It can be defined as follows: \(\cup F = \{ x \mid \exists A \in F (x \in A) \} = \cup_{i \in I} A_{i}\).
Properties
 Pairwise disjoint: A set family is pairwise disjoint if all its elements are disjoint: \(\forall X \in F \forall Y \in F (X \neq Y \implies X \cap Y = \emptyset)\)
Partitions
\(F\) is a partition of \(A\) if:
 \(\cup F = A\)
 \(F\) is pairwise disjoint
 \(\forall X \in F (X \neq \emptyset)\)
Uniqueness Operator
This operation allows us to refer to a single unique element of a set that matches the given constraints. Notice that the result is undefined if the given constraints don’t result in one and only one element.
Given a finite set of different natural numbers \(X\), we can obtain the largest one as: \(\mu x \in X \mid \forall y \in X \{ x \neq y \implies x > y \}\). Similarly, given the set \(Person\), there is only one element that wrote The Lord of the Rings. Such element is: \(\mu p \in Person \mid “The Lord of the Rings” \in books(p)\).
We might also add a term part to map the resulting element. For example: \(John = \mu p \in Person \mid “The Lord of the Rings” \in books(p) \circ firstname(p) \).
Comprehensions
A set comprehension is a handy mathematical notation to build sets based on arbitrarily complex expressions. The general form is \(\{ x \in X \mid P(x) \}\), which defines the set of all elements of \(X\) where \(P\) holds.
We can add a term part to map the elements in the set. For example, given \(address(p)\) returns the address of person \(p\), \(\{ p \in People \mid male(p) \circ address(p) \}\) is the set of addresses of all the male people.
If a set comprehension has no term part, like \(\{ x \in X, y \in Y \mid P(x, y) \}\), then the type the elements in the result depends on the comprehension’s characteristic tuple. In this case, the set comprehension begins with \(x \in X, y \in Y\), so the characteristic tuple is \((x, y)\), which is of type \(X \times Y\).
Notice that the variables used in the characteristic tuple depend on the variables used inside the scope of the comprehension. If we have \(\{ p \in X, q \in Y \mid P(p, q) \}\), then the characteristic tuple would be the pair \((p, q)\).
Binary Relations
Sets of tuples where the elements in the tuple have a logical relationship. Tuples may be expressed as \((x, y)\), or as \(x \mapsto y\), known as maplet notation. Given \(R \subseteq A \times B\), we say \(A\) and \(B\) are the source and target sets of \(R\). If \(A\) and \(B\) are of the same type, then the relation is homogeneous; if they are different, the relation is heterogeneous.
The powerset of a binary relation can be expressed with a doubleheaded arrow: \(A \leftrightarrow B = \wp (A \times B)\).
Cartesian Product
This operation can create binary relations from two sets consisting of all the possible tuple combinations: \(A \times B = \{ (a, b) \mid a \in A \land b \in B \}\).
The cartesian product is distributive over \(\cap\) and \(\cup\): \(A \times (B \cap C) = (A \times B) \cap (A \times C)\) and \(A \times (B \cup C) = (A \times B) \cup (A \times C)\).
Also note the following equations with regards to \(\cap\) and \(\cup\): \((A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)\) and \((A \times B) \cup (C \times D) \subseteq (A \cup C) \times (B \cup D)\).
With regards to the empty set, a cartesian product of the empty set and any set is equal to the empty set: \(\forall A (A \times \emptyset = \emptyset)\).
Lifted Form
Given a relation \(P \subseteq X \times Y\), the lifted form of \(P\) is denoted \(\mathring{P} = P \cup \{ \perp \} \times Y^{\perp}\). This is simply the original relation plus the association of the undefined element with all the elements of the augmented target.
Operations
Given \(R \subseteq (A \times B)\), a relation from \(A\) to \(B\):

Domain: The domain represents the set of all the elements from the left side of the ordered pairs: \(Dom(R) = \{ a \in A \mid \exists b \in B ((a, b) \in R)\}\)

Range: The range represents the set of all the elements from the right side of the ordered pairs: \(Ran(R) = \{ b \in B \mid \exists a \in A ((a, b) \in R)\}\)

Inverse: The relation with the ordered pairs reversed: \(R^{1} = \{ (b, a) \in B \times A \mid (a, b) \in R \}\). This may also be expressed as \(R^{\tilde{}}\)

Composition: Represents the combination of two relations where a certain elements from the ordered pairs are included in both relations. Given \(R \subseteq A \times B\) and \(S \subseteq B \times C\), \(S \circ R = \{ (a, c) \in A \times C \mid \exists b \in B ((a, b) \in R \land (b, c) \in S) \}\). Basically, \((S \circ R)(x) = R(S(x))\)
Note the following equation about inversing a composition: \((S \circ R)^{1} = R^{1} \circ S^{1} \).
If a certain binary relation is homogeneous, then \(R^{n}\) represents the result of composing \(R\) with itself \(n\) number of times. For example, \(R^{3} = R \circ R \circ R\).
Homogeneous (Binary) Relations
A binary relation \(R\) over \(A\) and \(B\) where \(A = B\) is called an homogenous relation, or an endorelation over \(A\).
The identity binary relation on \(A\) is denoted as \(i_{A} = \{ (a, a) \mid a \in A \}\).
Properties
Homogenous relations may have one or more of the following properties:

Reflexive: \(R\) is reflexive on \(A\) if \(\forall x \in A((x, x) \in R)\). Or in terms of the identity relation: \(i_{A} \subseteq R\)

Irreflexive: \(R\) is irreflexive on \(A\) if \(\forall x \in A((x, x) \notin R)\)

Symmetric: \(R\) is symmetric on \(A\) if \( \forall x \in A \forall y \in A (xRy \implies yRx) \). Or in terms of inverses, if \(R = R^{1}\)

Antisymmetric: \(R\) is antisymmetric on \(A\) if \( \forall x \in A \forall y \in A ((xRy \land yRx) \implies x = y) \)

Asymmetric: \(R\) is asymmetric on \(A\) if \( \forall x \in A \forall y \in A ((x, y) \in R \implies (y, x) \notin R) \)
Note that if \(R\) is asymmetric, then its also antisymmetric.
 Transitive: \(R\) is transitive on \(A\) if \( \forall x \in A \forall y \in A \forall z \in A ((xRy \land yRz) \implies xRz) \). Or in terms of composition, if \(R \circ R \subseteq R\)
Equivalence Relations
Equivalence relations are binary relations that are also reflexive, symmetric, and transitive. They can be used to partition a set into equivalence classes. In this context, an ordered pair means that the pair of elements should be considered “equivalent.”
Consider \(R = \{ (a, a), (b, b), (b, c), (c, b), (c, c) \} \). This equivalence relation tells us that \(a = a\), \(b = b\), \(c = c\), and \(b = c\). Therefore we have the following equivalence classes:
 \([a] = \{ a \}\), given that only \(a\) is equivalent to \(a\)
 \([b] = \{ b, c \}\), given that \(b\) is equivalent to \(c\) and to itself
 \([c] = \{ b, c \}\), given that \(c\) is equivalent to \(b\) and to itself
Given \(R \subseteq A \times A\), we can obtain the equivalent class of \(x \in A\) as \([x]_{R} = \{ y \in A \mid (y, x) \in R \}\).
Notice an element is always in the set of its equivalent classes, since an element is always equal to itself: \(\forall x \in A (x \in [x]_{R})\).
The set of all equivalence classes of all the elements of \(A\) is called “\(A\) module \(R\)”, denoted \(A \mathbin{/} R = \{ [x]_R \mid x \in A \}\). The resulting set family is always a partition of \(A\).
Restrictions
A binary relation can be restricted on the domain, or on the range.
A restriction on the domain means that we only consider pairs where the leftside element is a member of a certain set. Given \(R = X \times Y\) and \(Z \subseteq X\), the domain restriction of \(R\) by \(Z\) equals \(\{ (x, y) \in R \mid x \in Z \}\). This can be noted as \(Z \triangleleft R\).
Similarly, a restriction on the range means that we only consider pairs where the rightside element is a member of a certain set. Given \(R = X \times Y\) and \(P \subseteq Y\), the range restriction of \(R\) by \(P\) equals \(\{ (x, y) \in R \mid y \in P \}\). This can be noted as \(R \triangleright P\).
We can use restrictions to exclude a list of pairs from a binary relation, which is called domain or range substraction (or antirestriction). Following the above examples, we can exclude all pairs whose leftside element is in \(Z\) by doing \((X \setminus Z) \triangleleft R\), or we can exclude all pairs whose rightside element is in \(P\) by doing \(R \triangleright (Y \setminus P)\). These restrictions can also be expressed as \(Z ⩤ R\) and \(R ⩥ P\), respectively.
Relational Image
The relation image of a binary relation \(R \subseteq A \times B\) by a set \(P \subseteq A\) is the range of the domain restriction of \(R\) by \(P\): \(R(\vert P \vert) = range(P \triangleleft R)\). In other words, the relation image would be the set of all the results of function \(R\) where the argument is a subset of \(P\).
Closures
The closure of a relation \(R\) is the smallest relation containing \(R\) that satisfies a certain propery. For example, if \(R\) is an homogeneous relation, then \(R^{r} = R \cup i_{R}\) represents the reflexive closure of \(R\). If \(R = R^{r}\), then we say \(R\) is its own reflexive closure. Similarly, we can represent the symmetric closure of \(R\) as \(R^{s} = R \cup R^{1}\)
The transitive closure of an homogeneous relation is determined by composing the relation with itself enough times until the final relation stops growing. The union of all finite iterations of \(R\) is formally defined as \(R^{+} = \cup\{ n \in \mathbb{N} \mid n \geq 0 \circ R^{n} \}\), which is the transitive closure of \(R\), the smallest transitive relation containing \(R\).
For example, consider \(R = \{ (a, b), (b, c), (c, d) \}\). \(R^{1} = R\), \(R^{2} = \{ (a, b), (b, c), (b, d), (a, c), (c, d) \}\), and \(R^{3} = \{ (a, b), (b, c), (a, d), (b, d), (a, c), (c, d) \}\). But at this point, \(R^{4} = R^{3}\), \(R^{5} = R^{3}\), and so on. Therefore, \(R^{+} = R^{3}\), which is the transitive closure of \(R\).
Finally, the reflexive transitive closure of an homogeneous relation \(R\) is defined as \(R^{*} = R^{+} \cup i_{R}\).
Given a set, such as \(\mathbb{Z}\), and an operation, such as multiplication, we say that a set is closed under an operation if for all elements \(x\) and \(y\) from the set, the result of the operation (such as \(x * y\)), is already a member of the set.