Unit 1 Logic & Proofs

Question 1

What is the domain in the context of functions?

Answer 1

The set of all possible inputs.

Domain is crucial for understanding how functions operate.

Question 2

What is the codomain in the context of functions?

Answer 2

The set of all possible outputs.

Codomain is like the universe set for possible outputs of a function.

Question 3

What is the range of a function?

Answer 3

The set of all actual outputs that the function produces.

Question 4

How is a conditional statement structured?

Answer 4

If p, then q.

This structure is often represented as p → q.

Question 5

What does p represent in the statement ‘if p, then q’?

Answer 5

The hypothesis (the ‘if’ part).

Question 6

What does q represent in the statement ‘if p, then q’?

Answer 6

The conclusion (the ‘then’ part).

Question 7

What is a proof?

Answer 7

A step-by-step logical argument that shows a statement is true.

Question 8

What is a theorem?

Answer 8

The statement we are trying to prove.

Question 9

What is a direct proof?

Answer 9

The most straightforward type of proof used to prove conditional statements.

Question 10

What is the strategy for a direct proof?

Answer 10

Assume p is true, use definitions and logic to show q is true.

Question 11

What theorem is illustrated in the direct proof example?

Answer 11

If n is an even integer, then n² is also an even integer.

Question 12

What is the definition of an even integer?

Answer 12

An integer that can be written as 2 * k, where k is another integer.

Question 13

How do you show that n² is even if n is even?

Answer 13

By squaring the equation n = 2k to get n² = 4k².

Question 14

What is proof by contraposition?

Answer 14

Proving that ‘if not q, then not p’ is true.

Question 15

What is proof by contradiction?

Answer 15

Assuming both P and not Q are true to derive a contradiction.

Question 16

What is a tautology?

Answer 16

A logical expression that is always true.

Question 17

What is a contradiction?

Answer 17

A logical expression that is always false.

Question 18

What is a prime number?

Answer 18

A natural number greater than 1 with no positive divisors other than 1 and itself.

Question 19

What is a composite number?

Answer 19

A whole number greater than 1 that has more than two positive factors.

Question 20

What are real numbers?

Answer 20

All numbers that can be represented on a continuous number line, including rational and irrational numbers.

Question 21

What is a perfect square?

Answer 21

A number that can be expressed as the product of an integer with itself.

Question 22

What is an integer multiple?

Answer 22

A product of a number and any integer.

Question 23

How do you define consecutive integers?

Answer 23

If a is an integer, then b = a + 1.

Question 24

When is an integer x considered even?

Answer 24

If there is an integer k such that x = 2k.

Question 25

When is an integer x considered odd?

Answer 25

If there is an integer k such that x = 2k + 1.

Question 26

What does it mean for two numbers to have the same parity?

Answer 26

Both numbers are either even or both are odd.

Question 27

What is the definition of rational numbers?

Answer 27

Numbers that can be expressed as a fraction of two integers, where the denominator is not zero.

Question 28

Is 0 a rational number?

Answer 28

No. ## Footnote 0 can be expressed as 0/1, but it is not considered rational in this context.

Question 29

What does it mean for an integer x to divide another integer y?

Answer 29

x divides y if there exists an integer k such that y = k * x.

Question 30

What notation is used to denote that x divides y?

Answer 30

x | y.

Question 31

What does x ∤ y mean?

Answer 31

x does not divide y.

Question 32

What is the significance of a truth table?

Answer 32

It helps determine whether a logical expression is a tautology, contradiction, or neither.

Question 33

What are natural numbers?

Answer 33

Counting numbers (1, 2, 3, ...) ## Footnote Natural numbers are the set of positive integers used for counting.

Question 34

Define whole numbers.

Answer 34

Natural numbers plus zero (0, 1, 2, 3, ...) ## Footnote Whole numbers include all natural numbers and the number zero.

Question 35

What are integers?

Answer 35

Whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3, ...) ## Footnote Integers include all positive and negative whole numbers.

Question 36

How are rational numbers defined?

Answer 36

Numbers that can be expressed as a fraction of two integers (e.g., 1/2, -3/4, 0.75) ## Footnote Rational numbers can be written as a ratio of integers.

Question 37

What is a perfect square?

Answer 37

A number that can be expressed as the product of an integer with itself ## Footnote Example: 9 is a perfect square because it is equal to 3^2.

Question 38

Define integer multiple.

Answer 38

A product of a number and any integer ## Footnote Example: 12 is a multiple of 3 because 12 = 3×4.

Question 39

What does M(x) signify in the context of integer multiples?

Answer 39

x is an integer multiple of 4 ## Footnote Example: When x=8, n is 2 because 8 = 4×2.

Question 40

What are consecutive numbers?

Answer 40

Numbers that run continuously or in a logical sequence ## Footnote If a is an integer, then b=a+1.

Question 41

What are real numbers?

Answer 41

All numbers that can be represented on a continuous number line ## Footnote Real numbers include rational and irrational numbers.

Question 42

What is the base case in mathematical induction?

Answer 42

Verify the statement for the initial value (often k = 1) ## Footnote The base case establishes the starting point for induction.

Question 43

What is the inductive hypothesis?

Answer 43

Assume the statement is true for some arbitrary k = n ## Footnote This assumption is crucial for proving the inductive step.

Question 44

Define the inductive step in mathematical induction.

Answer 44

Prove it's true for k = n + 1 based on the assumption for k = n ## Footnote This step completes the induction process.

Question 45

What is proof by contradiction?

Answer 45

Assuming the negation of the statement you want to prove, then deriving a contradiction ## Footnote This method shows that the original statement must be true.

Question 46

What does proof by cases involve?

Answer 46

Dividing the problem into different cases that cover all possibilities ## Footnote Each case is proven separately.

Question 47

What is a tautology?

Answer 47

A logical expression that is always true, regardless of the truth values of its components ## Footnote Truth tables can be used to verify tautologies.

Question 48

What is proof by exhaustion?

Answer 48

Checks every element in the domain ## Footnote For example, checking each even number from 6 to 40 for Goldbach's conjecture.

Question 49

What is a Goldbach number?

Answer 49

A positive even integer that can be expressed as the sum of two odd primes ## Footnote Example: Prove that every even number from 6 to 40 is the sum of two odd primes.

Question 50

What is the contrapositive of a statement?

Answer 50

If P then Q becomes If not Q then not P ## Footnote A statement and its contrapositive are logically equivalent.