Set Theory and Functions

Question 1

|A| = k → |P (A)| = ?

P Denotes the Powerset

Answer 1

2ᵏ

Question 2

Let A₁ = {1,2,3}, A₂ = {1,2,4}, A₃ = {3,4,5}

A₁ ⋃ A₂ =

Answer 2

A₁ ⋃ A₂ = {1,2,3,4}

Question 3

Let A₁ = {1,2,3}, A₂ = {1,2,4}, A₃ = {3,4,5}

A₁ ⋂ A₂ =

Answer 3

A₁ ⋂ A₂ = {1,2}

Question 4

Let A₁ = {1,2,3}, A₂ = {1,2,4}, A₃ = {3,4,5}

Answer 4

A₁ ⋃ A₂ ⋃ A₃ = {1,2,3,4,5}

Question 5

What is the Powerset of a set, S?

Answer 5

The set of all subsets of S.
Always includes the empty set.

Question 6

Give an example of some distinct partitions of S.

S = {a,b,c,d}

Answer 6

{{},S}
{{a},{b,c,d}}
{{a},{b},{c},{d}}
{{a,b},{c,d}}
etc…

Question 7

Define a partition.

Answer 7

A finite or infinite collection of non-empty sets, the union of which have complete, distinct coverage of a set.

A = {A₁ ⋃ A₂ ⋃ A₃}

Question 8

Let A be a set. What is Aᶜ

Answer 8

The Compliment of A

Say A is a subset of X. Then Aᶜ = X - A.

It’s what is not in A.

Question 9

Difference between ⊆ and ⊂

Answer 9

⊆ = Subset - A set S is a subset of a set T, if every element that is in is S also in T.

⊂ = Proper Subset - A set S is a proper subset of a set T, if S ⊆ T and there is an element in T that is not in S.

For Sets A and B, Set A is a Subset of Set B if every element in Set A is also in Set B. It is written as ⊆ .

Proper Subsets - For Sets A and B, Set A is a Proper Subset of Set B if every element in Set A is also in Set B, but Set A does not equal Set B. ( ≠ ) It is written as ⊂

Question 10

Let A₁ = {1,2}, A₂ = {3,4,5}, A₃ = {6}

What is A₁ x A₂ x A₃

Answer 10

A₁ x A₂ x A₃ = {(1,3,6), (1,4,6), (1,5,6), (2,3,6), (2,4,6), (2,5,6)}

Question 11

f : X → Y

What is the Domain, Codomain and Image?

Also, what do they mean.

Answer 11

X = Domain: The set of all possible input values for a function.
Y = Codomain: The type of value that could come out of a function.
Image = The outputs a particular function actually uses is the image, also sometimes called the range.

Question 12

f : X → Y and B ⊆ Y

What is the pre image of B?

Also known as the inverse.

Answer 12

The “pre-image” (or “inverse image”) of a set under a function refers to the set of all input values that map to a given set of output values under that function.

More formally, given f : X → Y and B ⊆ Y the pre-image of B under f is the set of all elements x in A such that f(x) is in B.

Question 13

What is a One-to-One Function?

Also, what is it also known by?

Answer 13

A function f : A → B is one-to-one (or injective) if every element of A maps to a distinct element of B. In other words, if f (x₁) = f(x₂), then x₁ = x₂

This definition ensures that no two different inputs will produce the same output in a one-to-one function.

Also known as an Injective function

Question 14

Composition of functions:
Let f :X → Y and g : Y → Z be functions.

(g ∘ f)(x) = ?

Answer 14

g(f(x)) for all x ∈ X

Question 15

What is a Onto Function?

Also, what is it also known by?

Answer 15

A function f : A → B is onto (or surjective) if every element of B is the image of at least one element of A. In other words, for every b ∈ B, there exists an a ∈ A such that f (a) = b

This definition ensures that every possible output in the codomain B is produced by some input from the domain A in an onto function.

Also known as an Surjective function

Question 16

Define a Bijective function.

Answer 16

A function that is both one-to-one and onto

It also will have an inverse.