Relations and Functions – Key Vocabulary

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24 vocabulary flashcards covering fundamental terms from the chapter on relations and functions.

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24 Terms

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Relation

Any subset of a Cartesian product A × B; written a R b when (a,b) belongs to the relation

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Function

A special relation from X to Y assigning exactly one image in Y to every element of X

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Domain

The set of all first components (inputs) for which a function is defined

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Co-domain

The set Y that contains all possible outputs of a function f : X → Y

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Range

The actual set of images {f(x) | x ∈ X}; a subset of the co-domain

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Empty Relation

A relation R in A with R = ∅; no element is related to any other element

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Universal Relation

A relation R in A with R = A × A; every element is related to every element

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Reflexive Relation

A relation R on A in which (a,a) ∈ R for every a ∈ A

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Symmetric Relation

A relation R on A in which (a,b) ∈ R implies (b,a) ∈ R

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Transitive Relation

A relation R on A in which (a,b) and (b,c) in R imply (a,c) ∈ R

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Equivalence Relation

A relation that is reflexive, symmetric, and transitive

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Equivalence Class

For a relation R and element a, the set [a] = {x ∈ A | x R a}

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Partition of a Set

A collection of non-empty, pairwise disjoint subsets whose union equals the whole set; produced by an equivalence relation

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Injective (One-one) Function

A function f where f(x₁) = f(x₂) forces x₁ = x₂; images of distinct inputs are distinct

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Surjective (Onto) Function

A function f : X → Y where every y ∈ Y is f(x) for some x ∈ X

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Bijective Function

A function that is both injective and surjective

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Many-one Function

A function that is not injective; two or more domain elements share the same image

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Composition of Functions

For f : A → B and g : B → C, (g∘f)(x) = g(f(x)) mapping A directly to C

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Invertible Function

A bijective function f having an inverse f⁻¹ with f⁻¹∘f = IA and f∘f⁻¹ = IB

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Identity Function

IX : X → X defined by IX(x) = x for every x ∈ X

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Raster Method

A graphical/table method for displaying which ordered pairs belong to a relation

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Set-builder Method

Describing a set by a property, e.g., R = {(a,b) | b = a + 1}

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Finite-Set Mapping Property

For a finite set X, a function f : X → X is injective iff it is surjective (and vice-versa)

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Binary Operation

A rule that combines any two elements of a set to produce another element of the same set