Functions/Sets

Question 1

Preimage

Answer 1

‘a’ is a preimage of f(a)

Question 2

Domain

Answer 2

If f(x) maps A to B, A is the domain of function f(x).

Question 3

Codomain

Answer 3

If f(x) maps A to B, B is the codomain of function f(x).

Question 4

Image

Answer 4

If f(a) is b, b is the image of ‘a’ under function f(x).

Question 5

Range

Answer 5

The range of f(x) is the set of all images for elements in A.

Question 6

Function equality

Answer 6

Two functions are equal if:
- they have the same domain
- they have the same codomain
- they map the same element of domain to the same element of codomain

Question 7

Injective

Answer 7

One to one:
Each element in domain is mapped to at most one element in codomain

Question 8

Surjective

Answer 8

For each element in B, there exists an element in A such that f(a) = b.

Question 9

Bijective

Answer 9

Both Injective and Surjective;
both one to one and onto.
There is no element in B that is not mapped to from A.

Question 10

Composition

Answer 10

ie: f(g(x))

Question 11

Countably infinite set

Answer 11

A set that has a Bijection
(One-to-One Correspondence) with the Z domain

Question 12

Countable set

Answer 12

A set that is either Finite OR a set
that is Countably Infinite.

Question 13

Uncountable set

Answer 13

A set that is NOT a Countable
Set.

Question 14

Alphabet

Answer 14

A set of symbols

Question 15

String

Answer 15

A sequence of symbols based on an alphabet

Question 16

Language

Answer 16

A set of strings over some alphabet(s)

Question 17

Congruence Relation

‘a’ is congruent to ‘b mod m’ if:

Answer 17

m | a - b
m | a & m | b with same remainder

Question 18

IF
a ≡ b (mod m)
and c ≡ d (mod m)

Answer 18

THEN
a + c ≡ b + d (mod m)
a · c ≡ b · d (mod m)

Question 19

IF
a ≡ b (mod m) holds

Answer 19

THEN
c + a ≡ c + b (mod
m)

Question 20

TRUE or FALSE:
Dividing a valid congruence by an integer, c ∈ Z, preserves its validity

Answer 20

FALSE

Question 21

TRUE or FALSE:
Multiplying both sides of a valid congruence by an
integer, c ∈ Z, preserves its validity.

Answer 21

TRUE

Question 22

TRUE or FALSE:
Adding an integer to both sides of a valid congruence by an
integer, c ∈ Z, preserves its validity.

Answer 22

TRUE

Question 23

a + (b mod m) =

Answer 23

(a + b) mod m

Question 24

a x (b mod m) =

Answer 24

(a x b) mod m

Question 25

(a + (b mod m)) + (c mod m) =
(a x (b mod m)) x (c mod m) =

Answer 25

a + (b + (c mod m)) mod m)
a x (b x (c mod m)) mod m)

Question 26

a + (b mod m) =
a x (b mod m) =

Answer 26

b + (a mod m)
b x (a mod m)

Question 27

(a + (b mod m)) x (c mod m) =

Answer 27

(a x (c mod m)) + ((b x (c mod m)) mod m)

Question 28

a x ((b + (c mod m) mod m)

Answer 28

(a x (b mod m)) + ((a x (c mod m) mod m)