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24 vocabulary flashcards covering fundamental terms from the chapter on relations and functions.
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Relation
Any subset of a Cartesian product A × B; written a R b when (a,b) belongs to the relation
Function
A special relation from X to Y assigning exactly one image in Y to every element of X
Domain
The set of all first components (inputs) for which a function is defined
Co-domain
The set Y that contains all possible outputs of a function f : X → Y
Range
The actual set of images {f(x) | x ∈ X}; a subset of the co-domain
Empty Relation
A relation R in A with R = ∅; no element is related to any other element
Universal Relation
A relation R in A with R = A × A; every element is related to every element
Reflexive Relation
A relation R on A in which (a,a) ∈ R for every a ∈ A
Symmetric Relation
A relation R on A in which (a,b) ∈ R implies (b,a) ∈ R
Transitive Relation
A relation R on A in which (a,b) and (b,c) in R imply (a,c) ∈ R
Equivalence Relation
A relation that is reflexive, symmetric, and transitive
Equivalence Class
For a relation R and element a, the set [a] = {x ∈ A | x R a}
Partition of a Set
A collection of non-empty, pairwise disjoint subsets whose union equals the whole set; produced by an equivalence relation
Injective (One-one) Function
A function f where f(x₁) = f(x₂) forces x₁ = x₂; images of distinct inputs are distinct
Surjective (Onto) Function
A function f : X → Y where every y ∈ Y is f(x) for some x ∈ X
Bijective Function
A function that is both injective and surjective
Many-one Function
A function that is not injective; two or more domain elements share the same image
Composition of Functions
For f : A → B and g : B → C, (g∘f)(x) = g(f(x)) mapping A directly to C
Invertible Function
A bijective function f having an inverse f⁻¹ with f⁻¹∘f = IA and f∘f⁻¹ = IB
Identity Function
IX : X → X defined by IX(x) = x for every x ∈ X
Raster Method
A graphical/table method for displaying which ordered pairs belong to a relation
Set-builder Method
Describing a set by a property, e.g., R = {(a,b) | b = a + 1}
Finite-Set Mapping Property
For a finite set X, a function f : X → X is injective iff it is surjective (and vice-versa)
Binary Operation
A rule that combines any two elements of a set to produce another element of the same set