Week 3

Question 1

If given two sets, A and U, A is a subset of U. What is the compliment of A in U?

Answer 1

The compliment of A in U is a set that includes all the elements in U that are NOT in A.

Question 2

What are the De Morgan Laws when applied to compliments of sets?

Answer 2

A and B are subsets of U
(A union B) com in U = (A com in U) intersect (B com in U)
(A intersect B) com in U = (A com in U) union (B com in U)

Question 3

What is the Cartesian Product of A and B (A x B)?

Answer 3

The cartesian product is a set where each element of A x B has two parts: (a,b).
a comes from set A and b comes from set B

Question 4

What are the three components that define a function?

Answer 4

Domain - A collection of possible inputs
Codomain - A collection of possible outputs
Rule - A mathematical rule to apply

Question 5

What does it mean if two functions are equal?

Answer 5

-They have the same domain
-They have the same codomain
-Every output from the first function is equal to
the output from the second function

Question 6

What is the definition of an injective function?

Answer 6

Every element in the domain goes to a unique element in the codomain.

Question 7

What is the definition of a surjective function?

Answer 7

Every element in the codomain corresponds to an element in the domain.

Question 8

What is the definition of a bijective function?

Answer 8

A function that is injective and surjective.

Question 9

What does the following function composition mean:
f.g.h(x)

Answer 9

f(g(h(x)))

Question 10

Is function composition Associative?
(example f,g,h are functions)

Answer 10

Yes
h.(g.f) = (h.g).f

Question 11

What is the identity function and the symbol?

Answer 11

The identity function is a function when applied to a set will output the exact same set.
Symbol: idX(x) = x

Question 12

What does it mean for a function to have a left inverse?

Answer 12

• If we have two functions f and g
• f: X -> Y and g: Y -> X
  Starting at the domain (x) if we apply g.f(x) = x

Question 13

What does it mean for a function to have a right inverse?

Answer 13

• If we have two functions f and g
• f: X -> Y and g: Y -> X
  Starting at the domain (y) if we apply f.g(y) = y

Question 14

What does it mean for a function to be an inverse of another function?

Answer 14

It has both a left and right inverse.

Question 15

What can we conclude about a function that has a left inverse?

Answer 15

It is injective

Question 16

What can we conclude about a function that has a right inverse?

Answer 16

It is surjective

Question 17

What can we conclude about a function that has an inverse?

Answer 17

It is bijective

Question 18

If f is a function with a right inverse (g) and left inverse (h) then what can we conclude about g and h?

Answer 18

g=h