A function is injective (“one-to-one”) if \(x \neq x^{\prime} \Longrightarrow f(x) \neq f(x^{\prime})\) i.e. Given a set \(X\) and a set \(Y\), no two elements (say, \(x\) and \(x^{\prime}\)) from \(X\) map to the same element in \(Y\). (Note: a function can be both injective and surjective, this image is injective only)
If \(X\) and \(Y\) are sets, then we define the Cartesian Product \(X \times Y\) to be the collection of ordered pairs, e.g. \((x, y)\), whose first component lies in \(X\) and whose second component lies in \(Y\) \(X \times Y = \{(x, y) | x \in X, y \in Y\}\) An example would be \(\mathbb{R} \times \mathbb{R} = \mathbb{R}^{2}\) (i.e. the cartesian plane). \(\mathbb{R}^{2}\) is the cartesian product of “crossing” \(\mathbb{R}\) with itself. ...
Equality in Cardinality # Two sets \(X\) and \(Y\) are equal iff there exists a bijection \(f : X \longrightarrow Y\) from \(X\) to \(Y\).
A binary relation \(R\) over sets \(X\) and \(Y\) is a subset of the cartesian product of \(X \times Y\). Where \(X\) is the domain, or set of departure of \(R\), and \(Y\) is the codomain, or set of destination of \(R\). An element \(x \in X\) is related to \(y \in Y\), iff the ordered pair \((x, y)\) is found within the (above) subset.
A function is bijective or “invertible” if it is Both one-to-one and onto (injective and surjective) In other words: each element of one set is paired with exactly one element of the other set, and vice versa
Concepts # Relations Cardinality Binary Relation Symmetric Relation Transitive Relation Cartesian Product Reflexive Relation Surjective Function