D421 Discrete Math: Functions and Relations - Set 5 - Part 1

Correct Answer: 10

For disjoint sets there is no overlap, so the union has 6 + 4 = 10 elements.

Correct Answer: A

If A is a subset of B, then every element of A is in B, so their intersection is A itself.

Correct Answer: B

When A is contained in B, combining them adds nothing new to B, so the union equals B.

Correct Answer: Elements in exactly one of A or B

The symmetric difference contains elements in A or B but not in both, equal to (A − B) ∪ (B − A).

Correct Answer: True (vacuously)

A conditional with a false hypothesis is vacuously true regardless of the conclusion.

Correct Answer: 20

P(5, 2) = 5! / 3! = 5 × 4 = 20 ordered arrangements.

Correct Answer: 35

Order does not matter, so C(7, 3) = 7! / (3! × 4!) = 35.

Correct Answer: 1

By convention, 0! is defined to equal 1, which keeps combinatorial formulas consistent.

Correct Answer: Transitive

The three defining properties of an equivalence relation are reflexivity, symmetry, and transitivity.

Correct Answer: n

Congruence modulo n groups integers by their remainder, producing exactly n classes (0 through n − 1).

Correct Answer: Bijective

Only bijective functions are reversible, since each output corresponds to exactly one input and all outputs are used.

Correct Answer: Some cat is not black

Negating a universal statement gives an existential one: it is not the case that all are black means at least one is not black.

Correct Answer: No student passed

Negating an existential statement gives a universal one: it is false that some passed means every student did not pass.

Correct Answer: Modus ponens

Modus ponens is a valid rule of inference; affirming the consequent and denying the antecedent are logical fallacies.

Correct Answer: 6

K4 connects every pair of its 4 vertices, giving C(4, 2) = 6 edges.

Correct Answer: Has an isolated vertex

A vertex of degree 0 has no edges and is called an isolated vertex.

Correct Answer: 10

By the handshaking theorem, the degree sum equals twice the number of edges: 2 × 5 = 10.

Correct Answer: Their greatest common divisor is 1

Two integers are coprime when their only common positive divisor is 1, meaning GCD equals 1.

Correct Answer: Yes

The divisors of 8 are 1, 2, 4, 8 and of 15 are 1, 3, 5, 15; their only common divisor is 1, so they are coprime.

Correct Answer: 4

25 divided by 7 is 3 remainder 4, so 25 mod 7 = 4.