D421 Discrete Math: Functions and Relations - Set 1 - Part 3

Correct Answer: A set that can be put in bijection with the natural numbers

A countably infinite set has the same cardinality as the natural numbers, meaning its elements can be listed in a sequence.

Correct Answer: The real numbers

The real numbers are uncountable, as Cantor's diagonal argument shows no bijection with the naturals exists. The integers and rationals are countable.

Correct Answer: The set of outputs produced by applying the function to that subset

The image of a subset S under f is {f(x) : x in S}, the set of all outputs obtained from elements of S.

Correct Answer: The set of all inputs that map to y

The preimage of y is the set of all x in the domain such that f(x) = y.

Correct Answer: All diagonal entries are 1

In the 0-1 matrix of a relation, reflexivity requires every diagonal entry to be 1, since each element must relate to itself.

Correct Answer: The matrix equals its own transpose

A relation is symmetric exactly when its 0-1 matrix equals its transpose, since (a, b) present implies (b, a) present.

Correct Answer: Another name for a bijection

A one-to-one correspondence is a bijection, pairing each element of one set with exactly one element of another and vice versa.

Correct Answer: A to C

Composing g after f sends an element of A through f into B, then through g into C, so g ∘ f maps A to C.

Correct Answer: Strictly less than (<)

Strictly less than is not reflexive because no integer is less than itself. Equality, ≤, and divides are all reflexive.

Correct Answer: Each input produces exactly one unambiguous output

A function is well-defined when each input in the domain corresponds to exactly one output with no ambiguity.