Set Theory 2.4 & 2.5

Let A = {1, 2}, B = {a, b}. How many functions f: A→B are both total and surjective?

• 2

• 4

• 0

• 1

Let R⊆A×A be a relation such that R = {(a, b) ∣ a ≤ b}. Which of the following properties does R satisfy?

• Symmetric only

• Reflexive and transitive

• Antisymmetric and symmetric

• None of the choices

Given R= {(1,2),(2,3),(1,3)}, which properties does R satisfy?

• Reflexive only

• Transitive only

• Symmetric only

• None of the choices

Let f: Z→Z be defined as f(x) = x². Is this function injective?

• Yes

• No, because f(2) = f(−2)

• No, because it’s not surjective

• Yes, because each x has a unique image

How many binary relations on a set with 3 elements are symmetric?

• 2⁹

• 2⁶

• 2⁴

• 2³

Answer explanation

Explanation: 3 diagonal + 3 off-diagonal (only upper/lower triangle), 6 choices.

Let A = {x, y, z}. How many reflexive and symmetric relations are possible on A?

• 64

• 32

• 16

• 8

Answer explanation

Explanation: Diagonal fixed (3), choose presence/absence of 3 off-diagonal symmetric pairs: 2³ = 8, total: 2³ = 8.

Let f: R→R be defined by f(x) = 3x+1. Which of the following is true?

• f is not a function

• f is injective but not surjective

• f is bijective

• f is surjective but not injective

The composition of two injective functions is:

• Always injective

• Always surjective

• Not necessarily injective

• Identity

If f: A→B and g: B→C are both surjective, is g ∘ f necessarily surjective?

• Yes

• No

• Only if A = C

• Only if f is also injective

Suppose f(x) = (x)^1/2, with domain A = R. Why is this not a function?

• Square root is not defined for all x in A

• It’s not surjective

• The function gives two values

• Domain is not a set