C1-3

Question 1

What is the existential quantifier symbol?

Answer 1

3 for some

It indicates that there exists at least one element for which the statement is true.

Question 2

What does negate mean in logic?

Answer 2

to reverse the truth value of a statement

For example, negating a true statement makes it false.

Question 3

What is a tautology?

Answer 3

always true (i.e., PV TP)

A statement that is true in every possible interpretation.

Question 4

What is the complement rule in logic?

Answer 4

PV TP = T

This states that the disjunction of a proposition and its negation is always true.

Question 5

What does P^ -P = F signify?

Answer 5

a proposition and its negation cannot both be true

This is a fundamental principle in logic.

Question 6

What is the identity rule in logic?

Answer 6

P ∧ T = P

A proposition AND true is the proposition itself.

Question 7

What is the idempotent law?

Answer 7

P ∧ P = P

A proposition AND itself is the proposition.

Question 8

What does the commutative law state?

Answer 8

P ∧ Q = Q ∧ P

The order of propositions does not affect the outcome.

Question 9

What is the associative law?

Answer 9

(P ∧ Q) ∧ R = P ∧ (Q ∧ R)

The grouping of propositions does not affect the outcome.

Question 10

What is the distributive law?

Answer 10

P ∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R)

This shows how conjunction distributes over disjunction.

Question 11

What does double negation mean?

Answer 11

P = ¬(¬P)

Negating a negation returns the original proposition.

Question 12

What is De Morgan’s law?

Answer 12

¬(P ∧ Q) = ¬P ∨ ¬Q

It describes how negation interacts with conjunction and disjunction.

Question 13

What is the converse of P → Q?

Answer 13

Q → P

The converse switches the hypothesis and conclusion.

Question 14

What is the contrapositive of P → Q?

Answer 14

¬Q → ¬P

The contrapositive is logically equivalent to the original statement.

Question 15

What are the operations in algebra?

Answer 15

• • or •

These symbols represent multiplication in algebraic expressions.

Question 16

What is the identity property in algebra?

Answer 16

m + 0 = m and 0 + m = m

Adding zero to a number does not change its value.

Question 17

What does negation mean in algebra?

Answer 17

m + (-m) = 0

A number plus its negative equals zero.

Question 18

What is the commutative property in algebra?

Answer 18

m + n = n + m and m • n = n • m

The order of addition or multiplication does not affect the result.

Question 19

What is the associative property in algebra?

Answer 19

(m + n) + p = m + (n + p) and m • (n • p) = (m • n) • p

The grouping of numbers does not affect the result.

Question 20

What is the distributive property?

Answer 20

m • (n + p) = m • n + m • p

This shows how multiplication distributes over addition.

Question 21

What does m • n = 0 imply?

Answer 21

either m = 0 or n = 0

This is a property of multiplication in algebra.

Question 22

What is an order relation?

Answer 22

≤ (less than or equal to)

It defines how elements are compared in terms of size.

Question 23

What does reflexive mean in order relations?

Answer 23

every integer m satisfies m ≤ m

Each element is related to itself.

Question 24

What is the antisymmetric property?

Answer 24

if m ≤ n and n ≤ m, then m = n

This indicates that if two elements are mutually related, they are equal.

Question 25

What is the **transitive** property?

Answer 25

if m ≤ n and n ≤ p, then m ≤ p ## Footnote This shows a chain of relationships among elements.

Question 26

What does **comparability** mean?

Answer 26

given two integers, either m ≤ n or n ≤ m ## Footnote This indicates that any two elements can be compared.

Question 27

What is the **shift** property?

Answer 27

if m ≤ n, then m + s ≤ n + s ## Footnote Adding the same value to both sides preserves the inequality.

Question 28

What is the **scale** property?

Answer 28

if m ≤ n and 0 ≤ p, then m • p ≤ n • p ## Footnote Multiplying both sides by a positive number preserves the inequality.

Question 29

What is the **well ordering** principle?

Answer 29

any non-empty subset of Z* has a unique least element ## Footnote This principle applies to the set of positive integers.

Question 30

What does the **remainder theorem** state?

Answer 30

there is a unique q and r such that a = qb + r and 0 ≤ r < b ## Footnote This theorem describes the division of integers.

Question 31

What is the formula for **absolute value**?

Answer 31

|n| = * n if n ≥ 0 * -n if n < 0 ## Footnote Absolute value measures the distance of a number from zero.

Question 32

What is **Euclid's algorithm** used for?

Answer 32

to find the greatest common divisor (gcd) ## Footnote It involves subtracting the smaller number from the larger until reaching zero.

Question 33

What is **Bézout's identity**?

Answer 33

if a and b are nonzero, then there exist integers j and v such that gcd(a, b) = au + bu ## Footnote This identity relates the gcd to linear combinations of the integers.