7.3 Functions: Injective, Surjective, and Bijective

Function types are crucial in understanding how elements map between sets. Injective, surjective, and bijective functions each have unique properties that define their behavior and relationships between domain and codomain.

Identifying function types involves analyzing mapping behavior, using graphical representations, and examining domain and codomain relationships. Proofs for function classifications rely on specific techniques, while cardinality considerations play a key role in understanding set relationships.

Understanding Function Types

Types of function mappings

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• Injective (one-to-one) functions map distinct elements of domain to distinct elements of codomain ensuring each codomain element mapped to by at most one domain element ($x \neq y \implies f(x) \neq f(y)$) (linear function $f(x) = 2x$)

• Surjective (onto) functions ensure every codomain element mapped to by at least one domain element making range equal codomain ($\forall y \in Y, \exists x \in X : f(x) = y$) (sine function on $[-\pi/2, \pi/2]$)

• Bijective (one-to-one correspondence) functions combine injective and surjective properties establishing exact one-to-one correspondence between domain and codomain elements (exponential function $f(x) = e^x$)

Identification of function types

• Analyzing function behavior examines mapping between domain and codomain elements checking output uniqueness for injective and codomain coverage for surjective functions

• Graphical representation uses horizontal line test for injective functions ensuring every horizontal line intersects graph once for surjective functions

• Domain and codomain analysis compares cardinalities identifying restrictions or expansions affecting function type (restricting domain of $f(x) = x^2$ to non-negative reals makes it bijective)

Proofs for function classifications

• Injective proofs assume $f(x_1) = f(x_2)$ and show $x_1 = x_2$ or use contradiction assuming $x_1 \neq x_2$ to show $f(x_1) \neq f(x_2)$

• Surjective proofs start with arbitrary $y$ in codomain finding $x$ in domain where $f(x) = y$ using algebraic manipulation or function properties

• Bijectivity proved by demonstrating both injectivity and surjectivity separately or constructing inverse function

Cardinality in function mappings

• Injective functions preserve distinctness with domain cardinality less than or equal to codomain cardinality

• Surjective functions ensure all codomain elements "reached" with codomain cardinality less than or equal to domain cardinality

• Bijective functions establish one-to-one correspondence between sets with equal domain and codomain cardinality

• Finite sets require special considerations in determining function types while infinite sets have implications for countable and uncountable infinities (Cantor's diagonal argument)