Set Theory

Question 1

What is a group?

Answer 1

A combination of a set and an operation (e.g. addition over the integers) that satisfies all four group axioms: closure, associativity, identity element, inverse element

Question 2

What is the closure property?

Answer 2

For all x and y in G, x.y is also in G
∀ x, y ∈ G x.y ∈ G

Question 3

What is the associativity property?

Answer 3

∀ x, y, z ∈ G . (x.y).z = x.(y.z)

Question 4

What is the identity element?

Answer 4

There exists an element I, where x.I = I.x = x
∃ I ∈ G . ∀ x ∈ G . I.x = x.I = x

Question 5

What is the inverse element?

Answer 5

For all x in G, there exists an inverse of x named y, where x.y = y.x = I
∀ x ∈ G ∃ I ∈ G . x.y = y.x = I

Question 6

Meaning of ∃

Answer 6

there exists

Question 7

Meaning of ∀

Answer 7

for all/for any

Question 8

Meaning of .

Answer 8

such that

Question 9

Meaning of ∈

Answer 9

is an element of/belongs to

Question 10

What is meant by Z_n?

Answer 10

The set of integers modulo n (e.g. Z_4 = {0, 1, 2, 3})
2 + 3 = 1 (in Z it would be 5, but in Z_4 it’s 5 mod 4 = 1)

Question 11

What does A \ B mean? (where A and B are both sets)

Answer 11

The set of numbers which are in A but not in B
{e : e ∈ A, e !∈ B}

Question 12

What is a field?

Answer 12

A combination of a set and two operations, one with identity and an inverse, and another with identity and inverse for all except one element

Question 13

What is P(A)?

Answer 13

The set of all subsets of A

Question 14

What is A x B where A and B are sets?

Answer 14

Cartesian product of A and B, set of ordered pairs with first element from A and second element from B
A x B = {(a, b) . a ∈ A, b ∈ B}

Question 15

What is [n]?

Answer 15

The set of all integers from 1 to n

Question 16

What is an injective function?

Answer 16

A function f(x) where f(a) = f(b) implies a = b

Question 17

What is the image of a function?

Answer 17

The set of all output values it can produce

Question 18

What is the pre-image/inverse image of b under f(x)?

Answer 18

The set {a : f(a) = b}

Question 19

What is a surjective function?

Answer 19

Every output has an input that results in it (there can be more than one)

Question 20

What is a bijective function?

Answer 20

One which is injective and surjective

Question 21

When do two sets have the same size?

Answer 21

When there is a bijection between them

Question 22

How could you describe f(x) = sqrt(x)?

Answer 22

f : R>0 => R x R
f : R => C x C

Question 23

What is a commutative group?

Answer 23

A group in which it doesn’t matter which way round you write the elements when doing the operation because you get the same result
Addition under the real integers is a commutative group