D421 Discrete Math: Functions and Relations - Set 5 - Part 2

Correct Answer: The codomain of f matches the domain of g

For g ∘ f to be defined, the outputs of f must lie in the domain of g, so f's codomain matches g's domain.

Correct Answer: A itself

The identity function maps each element to itself, so its range is the entire set A.

Correct Answer: The empty relation

The empty relation contains no ordered pairs. It is vacuously symmetric and transitive but not reflexive on a non-empty set.

Correct Answer: True

Adding zero to any integer leaves it unchanged, so the universally quantified statement is true.

Correct Answer: 6

The number of 2-element subsets is C(4, 2) = 6.

Correct Answer: 1

The empty set has 2^0 = 1 subset, namely the empty set itself, so its power set has one element.

Correct Answer: It is always the empty set

Intersecting any set with the empty set yields the empty set, since there are no common elements.

Correct Answer: It is the set itself

Uniting a set with the empty set adds no elements, so the result is the original set.

Correct Answer: It is defined for every element of A

A total function assigns an output to every element of its domain A, unlike a partial function.

Correct Answer: Equal cardinality

A bijection pairs elements one to one, so the two finite sets must contain the same number of elements.

Correct Answer: f is a function from A to B

The notation f: A → B states that f is a function with domain A and codomain B.

Correct Answer: x is odd or x is not positive

By De Morgan's law, negating a conjunction gives a disjunction of negations: not even or not positive, i.e. odd or not positive.

Correct Answer: Difference between consecutive terms

An arithmetic sequence has a common difference added to each term to get the next.

Correct Answer: Ratio between consecutive terms

A geometric sequence multiplies each term by a common ratio to obtain the next term.

Correct Answer: 24

The common ratio is 2, so the 4th term is 12 × 2 = 24.

Correct Answer: 8

Each of the 3 inputs has 2 output choices, giving 2^3 = 8 functions.

Correct Answer: Distinct elements cannot both relate to each other

Antisymmetry forbids both (a, b) and (b, a) unless a = b, so two distinct elements cannot mutually relate.

Correct Answer: Is a sibling of

If A is a sibling of B, then B is a sibling of A, so the sibling relation is symmetric. Parent, older, and child relations are not.

Correct Answer: The original function

Composing any function with the identity leaves it unchanged, since the identity maps each element to itself.

Correct Answer: 6

CAT has 3 distinct letters, so the number of arrangements is 3! = 6.