1 speaking mathematically

Question 1

what are the 3 main kinds of mathematical statements?

Answer 1

1. universal statement
2. conditional statement
3. existential statement

Question 2

what is a universal statement?

Answer 2

• says that a certain property is true for all elements in a set
• e.g. all positive integers are greater than 0
• keywords: ‘all’, ‘every’, ‘any’
• symbol: ∀

Question 3

what is a conditional statement?

Answer 3

• says that if 1 thing is true then some other thing also has to be true
• e.g. if 378 is divisible by 18, then 378 is divisible by 6
• keyword: ‘if… then’
• symbol: –>

Question 4

what is an existential statement?

Answer 4

• says that there is at least 1 thing for which the property is true
• e.g. there is a prime number that is even
• keywords: ‘there exists’, ‘there is’, ‘some’
• symbol: ∃

Question 5

what is a universal conditional statement?

Answer 5

• both universal and conditional
• e.g. for all animals 𝑎, if 𝑎 is a dog, then 𝑎 is a mammal
• can be rewritten to appear purely universal or purely conditional

Question 6

what is a universal existential statement?

Answer 6

• universal bc 1st part says that a certain property is true for all objects of a given type
• existential bc 2nd part asserts the existence of sth
• e.g. every real number has an additive inverse

Question 7

what is an existential universal statement?

Answer 7

• existential bc 1st part asserts that a certain object exists
• universal bc 2nd part says that the object satisfies a certain property for all things of a certain kind
• e.g. there is a positive integer that is less than or equal to every positive integer

Question 8

what is an axiom / postulate?

Answer 8

a statement that is assumed to be true without proof

Question 9

what is a theorem?

Answer 9

mathematical statement that is proved using rigorous mathematical reasoning; usually a major or important result

Question 10

what is a lemma?

Answer 10

a small theorem; minor result whose purpose is to help in proving a theorem

Question 11

what is a corollary?

Answer 11

a result that is a simple deduction from a theorem

Question 12

what is a conjecture?

Answer 12

a statement believed to be true but for which there is no proof yet

Question 13

how do you define rational and irrational numbers?

Answer 13

• a real number 𝑟 is rational iff it can be expressed as a quotient of two integers with a nonzero denominator
• ie 𝑟 is rational ⟺ ∃ 𝑎, 𝑏 ∈ Z s.t. 𝑟 = 𝑎 / 𝑏 and 𝑏 ≠ 0
• real number that is not rational is irrational

Question 14

what does it mean for a fraction to be in its lowest term?

Answer 14

• fraction: quotient of 2 integers with a nonzero denominator
• fraction 𝑎 / 𝑏 (where 𝑏 ≠ 0) is said to be in lowest terms if the largest integer that divides both 𝑎 and 𝑏 is 1