Comp 1311 Mathematics 1

Question 1

What is the cardinality of a set?

Answer 1

The number of elements within a set

Question 2

What is a predicate?

Answer 2

Something that is either true or false about a member of some universe (set)

Question 3

Which set operations are associative?

Answer 3

Union
Intersection

Question 4

Which set operations are commutative?

Answer 4

Union
Intersection

Question 5

T/F The difference set operation is associative and commutative?

Answer 5

False
It is neither

Question 6

When is a set a subset of another?

Answer 6

When every element of X is also a member of Y

Question 7

What is a proper subset?

Answer 7

When a subset is strictly larger of smaller than another

Question 8

When are two sets equal?

Answer 8

When they contain the same elements

Question 9

What is a powerset of a set?

Answer 9

The set of all subsets of the original set.

Question 10

What is {a} + {b}?

Answer 10

{(a,0),(b,1)}

Question 11

What is a function?

Answer 11

A mapping between a domain and codomain where exactly one element of the domain maps to one element of the codomain

Question 12

What is the range?

Answer 12

A subset of the codomain, for the element of the codomain that are mapped to by a function

Question 13

What is the formula for the symmetric difference?

Answer 13

A △ B = (A-B) U (B-A)

Question 14

What is the identity function?

Answer 14

Every value maps uniquely to itself in the codomain

Question 15

How is a function with multiple inputs handled?

Answer 15

The domain is the Cartesian product of two sets

Question 16

Is function composition commutative, associative, both or neither?

Answer 16

Just associative

Question 17

When is a function injective?

Answer 17

for all x, x’ in X, if f(x)=f(x’), then x = x’
Elements can map to the same value in the codomain (0 –> 0) but two elements cannot map back to the same element

Question 18

When is a function surjective?

Answer 18

If for all values in the codomain, there exists an element in the domain which maps to it.

Question 19

When is a function bijective?

Answer 19

When a function is both injective and surjective

Question 20

What is a relation from X to Y?

Answer 20

The set of some ordered pairs made from X and Y’s Cartesian product.

Question 21

What is the formula for negative binomial distribution

Answer 21

p(k|θ) = <nk>(1 − θ)^n θ^k</nk>

Question 22

What is the formula for geometric distribution?

Answer 22

p(k|θ) = (1 − θ)^(k−1) θ

Question 23

What is the expected value and variance for geometric distribution?

Answer 23

E[X] = 1/θ
V[X] = (1 - θ) / θ^2

Question 24

What is the formula for binomial distribution?

Answer 24

(nCk) θ^k (1 − θ)^(n-k)

Question 25

What is the formula for Bernoulli distribution?

Answer 25

θ^x (1 − θ)^(1−x)
θ ∈ [0, 1]

Question 26

What is the expected value formula for Bernoulli and Binomial?

Answer 26

Ber: E[X] = θ
Binomial: E[X] = nθ
E[X^2] = nθ + n(n-1)θ^2

Question 27

What is the variance formula for Binomial?

Answer 27

V[X]=nθ(1-θ)

Question 28

What is the expected value and variance of uniform distribution?

Answer 28

E[X] = (a+b)/2
V[X]=((b - a + 1)^2 - 1) / 12

Question 29

What is the formula for Poisson distribution?

Answer 29

P(X = k|λ) = (λ^k e^−λ) / k!

Question 30

What is the variance and expected value of Poisson distribution?

Answer 30

λ

Question 31

What is λ and k in Poisson distribution?

Answer 31

λ is the constant average rate of events
k is the number of events

Question 32

What is the covariance formula?

Answer 32

cov(X,Y) = E[XY]-E[X]E[Y]