MAT 1362 Midterm One

0.0(0)
studied byStudied by 0 people
learnLearn
examPractice Test
spaced repetitionSpaced Repetition
heart puzzleMatch
flashcardsFlashcards
Card Sorting

1/45

encourage image

There's no tags or description

Looks like no tags are added yet.

Study Analytics
Name
Mastery
Learn
Test
Matching
Spaced

No study sessions yet.

46 Terms

1
New cards

Definition of Axiom

Statement or proposition, accepted as true without proof.

2
New cards

Definition of Set

Collection of things labeled S, obtains elements/members

3
New cards

Binary Operation

Action on a set, they define a function. Typically depend on two or more elements.

4
New cards

Everything is connected to ______ and ______ in a set

addition, multiplication

5
New cards

Axiom 1.1 → Commutativity, associatory and distributivity

  1. a+b = b+a (C)

  2. (a+b)+c = a+(b+c) (A)

  3. a(b+c) = ab + ac (D)

  4. ab = ba (C)

  5. (ab)c = a(bc) (A)

6
New cards

Axiom 1.2 → Addition Identity

There exists an integer 0 s.t a+0 = afor all integers a.

7
New cards

Axiom 1.3 → Multiplicative

Identity There exists an integer 1 such that a×1 = a for all integers a.

8
New cards

Axiom 1.4 → Additive Inverse

For every integer a, there exists an integer -a such that a + (-a) = 0.

9
New cards

Axiom 1.5 → Cancellation

If a, b, and c are integers and a + b = a + c, then b = c.

10
New cards

Properties of =

  1. a = a (Reflexivity)

  2. a=b, then b=a (symmetry)

  3. a=b, b=c, then a=c (Transitivity)

  4. a=b, then a can be replaced by b in any statement or expression

11
New cards

Properties of Does not equal

  1. NOT reflexive

  2. Symmetric

  3. NOT transitive

12
New cards

How do you mark the end of a proof

with a square

13
New cards

Proposition 1.12 → (Uniqueness of the additive identity)

Suppose a ∈ Z. If a has the property that b + a = b for all b ∈ Z, then a = 0. In other words, the additive identity is unique.

14
New cards

What does ‘or’ mean in maths

In mathematics, 'or' refers to a logical disjunction where at least one of the statements must be true. It is often used in conditional statements and set theory to denote options or alternatives.

15
New cards

Definition 1.27 (Subtraction)

For a,b ∈ Z, we define a−b to be a+(−b). This new binary operation − will be called subtraction

16
New cards

Axiom 2.1 (Natural numbers)

There exists a subset N of Z with the following properties:

(i) If a,b ∈ N, then a+b ∈ N. (The subset N is closed under addition.)

(ii) If a,b ∈ N, then ab ∈ N. (The subset N is closed under multiplication.)

(iii) 0 ̸∈ N.

(iv) For every a ∈ Z, we have a ∈ N or a = 0 or −a ∈ N.

17
New cards

What is a natural number

A positive integer

18
New cards

What is a proof by contradiction

A method of mathematical proof where the negation of the statement to be proved is assumed, leading to a contradiction. This implies that the original statement must be true.

19
New cards

Not a subset

20
New cards

subset or equal subset

21
New cards

not a subset and not an equal set

22
New cards

strict subset

23
New cards

Set Notation

{nEZ : some property of n}

24
New cards

Axiom 2.17 (Induction Axiom).

Suppose a subset A ⊆ Z satisfies the following properties:

(i) 1 ∈ A

(ii) n ∈ A =⇒ n+1∈A.

Then N ⊆A

25
New cards

Theorem 2.19 (Principle of mathematical induction: first form).

Suppose that, for each k ∈N, we have a statement P(k). Furthermore, suppose that

(i) P(1) is true

(ii) for all n ∈ N, P(n) ⇒ P(n+1).

Then P(k) is true for all k ∈ N

26
New cards

What are the steps of induction

  1. Identify p(x)

  2. Prove base case (p(1))

  3. prove p(x+1), Remember to use induction hypothesis

27
New cards

Theorem 2.26 (Principle of mathematical induction: first form revisited)

Suppose m is a fixed integer and that, for each k ∈ Z with k ≥ m, we have a statement P(k). Furthermore, suppose that (i) P(m) is true, and (ii) for all n ≥ m, P(n) =⇒ P(n+1). Then P(k) is true for all k ≥ m

28
New cards

Definition 2.28 (Smallest and greatest elements).

Suppose A ⊆ Z is nonempty. If there exists m ∈ A such that m≤afor all a ∈ A, then we say m is a smallest element of A and write m = min(A). If there exists M ∈ A such that M ≥afor all a ∈ A, then we say M is a greatest element of A and write M = max(A)

29
New cards

Theorem 2.33 (Well-ordering principle).

Every nonempty subset of N has a smallest element.

30
New cards

Definition 2.36 (gcd).

Suppose a,b ∈ Z. If a and b are not both zero, we define gcd(a,b) = min {k ∈ N : k = ax+by for some x,y ∈ Z} .

31
New cards

for all

32
New cards

there exists

33
New cards

(∃0 ∈ Z such that)(∀a ∈ Z)

Quantified Statement

34
New cards

Does not exist

35
New cards

∃!

There exists a unique

36
New cards

Implies

37
New cards

⇐⇒

If and only if

38
New cards

What is the converse

Reversing the Statements, for example p implies q becomes q implies p.

39
New cards

What is the contrapositive

Negating the expressions and reversing, creates equivalent statements for example P implies Q becomes not Q → not P

40
New cards

what is the negation of ‘and’

or

41
New cards

What is the negation of ‘or’

and

42
New cards

¬

negation (or not)

43
New cards

What is De Morgans laws

¬(A ∧ B) is equivalent to (¬A ∨ ¬B), and ¬(A ∨ B) is equivalent to (¬A ∧ ¬B).

44
New cards

and

45
New cards

or

46
New cards

How to negate quantifiers

  1. Same place, swap symbol (for all → exists)

  2. Negate final statement