Math 163

Question 1

Mapping Function

Answer 1

If we have two sets one A and one B if A -> (maps) to B. Every element in A must map to another element. And elements in A cannot map to another element more than once.

Question 2

What is a function?

Answer 2

(input - output) relation also for every element in the domain there is exactly one element in the codmain (ordered pair)

Question 3

What is an image?

Answer 3

If we have a a function going from A to B we say that in (a,b) b is the image/output of a

Question 4

What is a pre-image?

Answer 4

If we have a a function going from A to B we say that in (a,b) a is the pre-image/input of b

Question 5

Surjective

Answer 5

Let A and B be sets and f is function from A to B. It is surjective when at least all of the elements in A map to all of the elements in B.

Question 6

Injective

Answer 6

Let A and B be sets and f is a function from A to B. We say f is injective when every element of A has a unique image.

Question 7

Bijective

Answer 7

When the function is both injective and surjective

Question 8

Inverse

Answer 8

(Will come back to)

Question 9

Relation

Answer 9

Is a subset of AxB

Question 10

Partial Order

Answer 10

Is a relation that has the properties of reflexive, antisymmetric, and transitive.

Question 11

What properties does a relation need when it is on a graph set?

Answer 11

antireflexive and symmetric

Question 12

What does a | b mean?

Answer 12

It means that b = a * k where k is a natural number. b is some multiple of a.

Question 13

Equivalence Relation

Answer 13

Must be reflexive, symmetric, and transitive.

Question 14

What does S \ R mean

Answer 14

All of the elements in S that are that are in not is R.

Question 15

Mathematical Statement

Answer 15

A declarative true or false statement cannot be both.

Question 16

Conjunction(∧)

Answer 16

Whenever T(P) = 1 and T(Q) = 1
then, T(P ∧ Q) = 1. Otherwise T(P ∧ Q) = 0

Question 17

Disjunction(∨)

Answer 17

Whenever T(P) = 0 and T(Q) = 0
then, T(P ∨ Q) = 0. Otherwise T(P ∨ Q) = 1

Question 18

Negation(¬)

Answer 18

Means not P or that statement

Question 19

Mathmatical Formula

Answer 19

Is a statement that has a truth value for each element in a set.

Question 20

Universal Quantifier

Answer 20

“For every”

Question 21

Existential Quantifier

Answer 21

“There exists” or “At least one”

Question 22

Principle of mathematical induction

Answer 22

It is a template for constructing a mathmatical proof that consists of two main components. The base case and the inductive step.

Question 23

What is the outline of mathematical induction?

Answer 23

1.) State Fact
2.) Prove base case
3.) Assume P(k) to be true to prove P(k+1) to be true
4.) Express that your claim is valid for your range of elements

Question 24

What does this symbol mean “^”

Answer 24

Called a “conjunction”. This means when T(P) = 1 and T(Q) = 1 then it’s true else it’s false(zero).

Question 25

What does this symbol mean "∨"

Answer 25

Called a "disjunction". This means when T(P) = 0 and T(Q) = 0 then it's false(zero) else it's true.

Question 26

What does this symbol mean "⇒"

Answer 26

Called an "implication". This means that If P, then Q, The only time this is false is when P is true and Q is false. Think of P as the hypothesis and Q as the conclusion

Question 27

What does this symbol mean "↔"

Answer 27

It's called "biconditional". If A ⇒ B and B ⇒ A have the same truth value then it is true else it is false.

Question 28

Converse

Answer 28

Reversing the statement. If the original statement was P ⇒ Q then it would be Q ⇒ P.

Question 29

Contrapositive

Answer 29

Reversing the statement rather than negating both statements. This is equal to the same truth values as the original statements. ¬ Q ⇒ ¬ P.

Question 30

Define this symbol "ℂ"

Answer 30

It is called the complex number. Defined as {a+b𝒾 | a,b ∈ R}, Where "a" is the real part and "b" is the imaginary part.

Question 31

Imaginary Unit "𝒾"

Answer 31

It means the number which is a solution of x^2 + 1 = 0. Morover 𝒾 = square root (-1).