An object \(x\) is any abstract entity that can be formally defined and assigned to a variable.

A set \(X\) is an unordered collection of distinct objects. Sets are themselves objects.

If an object \(x\) lies in the set \(X\) then \(x\) is an element of \(X,\) \(x \in X.\) Otherwise, \(x\) is not an element of \(X,\) \(x \notin X.\)

Two sets \(X\) and \(Y\) are equal, \(X=Y,\) if and only if every element of \(X\) is an element of \(Y\) and every element of \(Y\) is an element of \(X.\) Otherwise \(X\) and \(Y\) are not equal, \(X \neq Y.\)

A set containing no elements is the empty set \(\varnothing .\) A singleton set is the set \(\{x\}\) whose only element is \(x.\) A pair set is the set \(\{x, y\}\) whose only elements are \(x\) and \(y.\)

A set \(X\) is a subset of another set \(Y,\) \(X \subseteq Y,\) if and only if every element of \(X\) is also an element of \(Y.\) \(X\) is a proper subset of \(Y,\) \(X \subset Y\) if and only if \(X \subseteq Y\) and \(X \neq Y.\)

Let \(X\) be a set. \(P(x)\) is a predicate if it depends on each \(x \in X\) and evaluates to a true or false statement.

Let \(X\) be a set and \(P(x)\) be a predicate depending on \(x \in X.\) If \(P(x)\) is true for all \(x \in X,\) the universal quantifier \(\forall\) can be used to denote “for all”.

\[\forall x \in X : P(x)\]

means “the predicate \(P(x)\) is true for all elements in \(X\)”.

If \(P(x)\) is true for some \(x \in X,\) the existential quantifier \(\exists\) can be used to mean “there exists some”.

\[\exists x \in X : P(x)\]

means “there exists some elements in \(X\) such that the predicate \(P(x)\) is true”.

Let \(X\) be a set and for each element \(x \in X\) let \(P(x)\) be a predicate depending on \(x.\) Then there exists a set \(\{x \in X : P(x)\}\) whose elements are the elements in \(X\) for which \(P(x)\) is true where the symbol “:” means “such that”. Hence \(\{x \in X : P(x)\}\) means “the set of elements in \(X\) such that the predicate \(P(x)\) is true”. Building sets from predicates is called specification.

The union of a set \(X\) with another set \(Y,\) \(X \cup Y,\) is the set of elements in \(X,\) \(Y\) or both.

\[X \cup Y := \{x : x \in X \ \mathrm{or} \ x \in Y\}\]

The intersection of a set \(X\) with another set \(Y,\) \(X \cap Y,\) is the set of elements in both \(X\) and \(Y.\)

\[X \cap Y := \{x : x \in X \ \mathrm{and} \ x \in Y\}\]

The relative complement of a set \(X\) in another set \(Y,\) \(Y \setminus X\) is the set of elements in \(Y\) that are not in \(X.\)

\[Y \setminus X := \{x : x \in Y \ \mathrm{and} \ x \notin X\}\]

The symmetric difference of a set \(X\) with another set \(Y,\) \(X \ominus Y\) is the set of elements in \(X\) or \(Y\) and not in \(X\) and \(Y.\)

\[X \ominus Y := \{x : x \in X \ \mathrm{and} \ x \in Y \ \mathrm{and} \ x \notin X \cup Y\}\]

The Cartesian product of sets \(X\) and \(Y,\) \(X \times Y\) is the set of all ordered pairs \((x,y)\) where \(x\) is an element of \(X\) and \(y\) is an element of \(Y.\)

\[X \times Y := \{(x,y) : x \in X \ \mathrm{and} \ y \in Y\}\]

Two sets \(X\) and \(Y\) are disjoint if and only if their intersection is the empty set.

\[X\cap Y=\varnothing\]

The power set \(\mathcal{P}(X)\) of a set \(X\) is the set of all subsets of \(X.\)

\[\mathcal{P}(X) := \{A : A \subseteq X \}\]

A partition \(P\) of a set \(X\) is a set of non-empty subsets of \(X\) such that every element \(x\in X\) is in exactly one subset in \(P.\) Partitions of sets have the following properties:

\[\varnothing \notin P\] \[\bigcup_{A\in P}=X\] \[A\cap B = \varnothing\,\,\forall\,A,B\in P, A\neq B.\]

A set \(X\) has cardinality \(n \in \mathbb{N}\) if and only if it has equal cardinality with \(\{i \in \mathbb{N} : i < n\} .\) \(X\) has \(n\) elements if and only if it has cardinality \(n.\)

A set \(X\) is finite if and only if it has cardinality \(n \in \mathbb{N}.\) Otherwise, the set is infinite.

A set \(X\) is countable if and only if it has cardinality equal with the natural numbers \(\mathbb{N}.\) The set is uncountable if it is infinite and not countable.