Logic

Question 1

What is a predicate?

Answer 1

A predicate is a sentence that contains finitely many variables and becomes a statement if the variables are each assigned a value.

Question 2

What word defines the list of all possible values that may be assigned to each variable (in a predicate)

Answer 2

Domain

Question 3

True or false? The following is a predicate

“Is x and integer”

Answer 3

False.

When something is put into x, it will become a question, not a statement

Question 4

What is the meaning of a truth set of a predicate

Answer 4

The truth set, P(x), is the set of all values in the domain, that when assigned to x, will make P(x) true.

Example

Question 5

What does ℤ mean?

Answer 5

It defines the list of all posible integers

It is a common domain

Question 6

What does ℤ⁺ mean?

Answer 6

It defines the list of all posible positive integers

It is a common domain

Question 7

What does ℤ^nonneg mean?

Answer 7

It defines the list of all posible integers that are not negative including zero

It is a common domain

Question 8

What does ℕ mean?

Answer 8

The list of all natural numbers which means numbers that are greater or equal to 1

It is a common domain

Question 9

What does ℚ mean?

Answer 9

The list of all rational numbers. Meaning numbers that divide into an integer

It is a common domain

Question 10

What does ℝ mean?

Answer 10

The list of all real numbers

It is a common domain

Question 11

True or false?

Predicates can be written as follows: P(x,y) = x is a student at y

Answer 11

True

Question 12

Answer 12

Question 13

Answer 13

Question 14

What does the symbol ∀ mean and what is it called?

Answer 14

The universal quantifier and it means “for all”

Question 15

When is the statement ∀ x ∈D, Q(x) true?

Answer 15

If and only if, Q(x) is true for each individual x in D

Question 16

When is the statement ∀ x ∈D, Q(x) false?

Answer 16

If and only if, Q(x) is false for at least one x in D

Question 17

Let D = {1, 2, 3, 4, 5}, and consider the statement

∀ x ∈ D, x² ≥ x
Write one way to write this sentence in full english

(no symbols)

Answer 17

“For every x in the set D, x² is greater than or equal to x.
This statement is true

Question 18

Consider the statement

∀ x ∈ R, x² ≥ x
Find a counterexample to show that this statement is false

Answer 18

x=1/2

Question 19

What is the method of exhaustion?

Answer 19

It is when you try proving the truth of a predicate for each and every individual element of a domain

Question 20

What the does symbol ∃ define and what is it called?

Answer 20

It denotes “there exists” and is called the existential quantifier

Question 21

When is the statement ∃ x ∈D, Q(x) true?

Answer 21

If and only if, Q(x) is true for at least one x in D

Question 22

Rewrite the following formal statement sin a variety of equivalent but more informal ways.
a. ∀ x ∈ ℝ, x² ≥ 0
b. ∀ x ∈ ℝ, x² -1 ≠ 0
c. ∃m ∈ ℤ⁺ such that x² = m

(Do note use the the symbol ∃ or ∀)

Answer 22

Question 23

What is meant by a quantifier ‘trailing’ a statement

Answer 23

Instead of writing with a quantifer at the start of the sentence (“For any real number x, x² is nonnegative”), the quanitfer is changed to the end of the sentence (“x² is nonnegtaive for any real number x“)

Question 24

Answer 24

Question 25

Answer 25

Question 26

Answer 26

Question 27

Rewrite the folowing statement in the form ∀\_\_\_, if \_\_\_ then \_\_\_. "If a real number is an integer, then it is a rational number"

Answer 27

Question 28

Given a statement, "All bytes have eight bits", is it mathmatically correct to translate it to ∀x, if x is a bute, then x has eight bits (emitting the identification of the domain)

Answer 28

Yes this is mathmatically correct

Question 29

# True or false? In a universal condition statement (∀\_\_\_, if p then q.), if you know that the hypothesis (p) is false, you must still interpret it as true

Answer 29

True

Question 30

How would you define a prime number in english?

Answer 30

An integer that is greater than 1 whose only positive integer factors are itself and 1.

Question 31

# True or false? The statement "The number 24 can be written as a sum of two even integers" is a existential quantification

Answer 31

True as it can be expressed formally as "**∃** even integers, m and n, such that 24 = m+n"

Question 32

# What is the negation of the following statement?

Answer 32

Reverse the quantifer and negate Q(x)

Question 33

# What is the negation of the following statement?

Answer 33

Reverse the quantifer and negate Q(x)

Question 34

# Write formal negations for the following statements: a) ∀ primes p, p is odd b) ∃ a triangle T such that the sum of all the angles of T equals 200°

Answer 34

a) ∃ a prime p, such that p is not odd b) ∀ triangles T, the sum of all the angles of T do not equal 200°

Question 35

# Write the negation for the following statement: ∀ people p, if p is blonde then p has blue eyes

Answer 35

∃ a person p, such that p is blonde and p does not have blue eyes

Question 36

# True or false? ∀ is just a generalisation of an 'and' statement

Answer 36

True

Question 37

# True or false? ∃ is just a generalisation of an 'or' statement

Answer 37

True

Question 38

# True or false? Consider the statement: ∀ x in D, if P(x) then Q(x) If P(x) is false for all x in D, then statement is by default, true.

Answer 38

True. This is shown by the nature of if statements (if p is false, the statement will automatically/vacuously true)

Question 39

# True or false? The rules of conditional statements (contrapositive, inverse and converse) also apply to universal conditional statements | (∀ x in C, if p then q)

Answer 39

True

Question 40

# Write the formal contrapositive, converse and inverse of the following: If a real number is greater than 2, then its square is greater than 4.

Answer 40

Original formal: ∀ x ∈ ℝ, if x>2 then x²>4 Contrapositive: ∀ x ∈ ℝ, if x²≤4 then x≤2 Converse: ∀ x ∈ ℝ, if x²>4 then x>2 Inverse: ∀ x ∈ ℝ, if x≤2 then x²≤4

Question 41

Answer 41

Question 42

Answer 42

Question 43

The reciprocal of a real number, a, is a real number, b, such that ab = 1. The following statement is true. Rewrite it formally using quantifiers and variables: "Every nonzero real number has a reciprocal"

Answer 43

∀ nonzero real numbers, u, ∃ a real number v such that uv=1 **or** ∀ u ∈ ℝ^nonzero, ∃ v ∈ ℝ such that uv=1

Question 44

The reciprocal of a real number, a, is a real number, b, such that ab = 1. The following statement is true. Rewrite it formally using quantifiers and variables: "There is a real number with no reciprocal"

Answer 44

∃ a real number, u, such that ∀ real numbers, v uv=1 **or** ∃ u ∈ ℝ such that ∀ v ∈ ℝ uv=1

Question 45

# Consider the statemnet "There is a smallest positive integer" Write this statement formally using both ∃ and ∀

Answer 45

∀ m ∈ ℤ⁺, ∃ n ∈ ℤ⁺ such that n ≤ m ## Footnote Given two integers, n and m. There exists a postive integer (n) that is always less than or equal all the other integers (m)

Question 46

# Consider the statemnet "There is no smallest positive real number" Write this statement formally using both ∃ and ∀

Answer 46

∀ m ∈ ℝ⁺, ∃ n ∈ ℝ⁺ such that m > n ## Footnote Given two integers, n and m. There exists a postive integer (n) that is always less than or equal all the other integers (m)

Question 47

# True or false? With negations of statements with more than one qualifier, you can apply the same rules as one qualifier

Answer 47

True

Question 48

# True or false? You can never change the order of quanitifiers

Answer 48

False. It is true that when there are different quanitifers in a statement, changing their order will change the meaning but if there are two of the same quantifiers, chaning the order does not matter.