Discrete Exam 1 Review

Indicate whether the statement is true or false. Use the inclusive OR. 2 is a negative number or -3 is a positive number.

False

r: If it rains today, I will have my umbrella. It is raining today. I do not have my umbrella. True or False?

False

In the following question, the domain is the set of all real numbers. x⁢y means x times y. ∀x∃y⁡(x⁢y=1)

False. If the universal player selects x to be 0, then there is no value that the existential player can select for y that can make x⁢y=1.

In the following question, the domain is the set of all nonnegative integers. x⁢y means x times y. ∀x∀y⁡(x⁢y=1)

False. counterexample: x =1 & y = 2.

Determine the truth value of the proposition. Then write down the negation of the proposition. ∃x∈R, ∀y∈R, y^2 < x + 1

False. ∀x∈R, ∃y∈R, y^2 >= x + 1.

Determine the truth value of the proposition. Then write down the negation of each proposition. ∀ integer n, n^2 is odd and n^3 is even.

False. ∃ integer n, n^2 is even or n^3 is odd.

Is the following conditional statement true? If January has 31 days, then 7 is an even number.

False. The hypothesis is true, but the conclusion is false.

Indicate whether the universally quantified statement is true or false. ∀x⁡(x − 1 > 0)

False; counterexample: x = 1

Establishing that ∃x⁢A⁡(x) is true

Finding an element in the domain for which the predicate is true

Given a proposition: if 3 + 5 < 2, then 1 + 3 =/= 4. Write down the converse of the proposition and decide the truth value of the converse.

If 1 + 3 =/= 4, then 3 + 5 < 2. True

Given a proposition: if 3 + 5 < 2, then 1 + 3 =/= 4. Write down the contrapositive of the proposition and decide its truth value.

If 1 + 3 =5, then 3 + 5 >= 2. True.

Ex. If it is raining today, the game will be cancelled. What is the inverse?

If it is not raining today, the game will not be cancelled.

opposite parity

If one number is odd and the other is even

Ex. If it is raining today, the game will be cancelled. What is the converse?

If the game is cancelled, it is raining today.

Ex. If it is raining today, the game will be cancelled. What is the contrapositive?

If the game is not cancelled, then it is not raining today.

same parity

If two numbers are both even or both odd

If x divides y, then y is said to be a __ of x

If x divides y, then y is said to be a multiple of x

State the inverse, contrapositive, and converse of the conditional statement. Then indicate whether the inverse, contrapositive, and converse is true: If 7 < 5, then 5 < 3.

Inverse: If 7 ≥ 5, then 5 ≥ 3. (True). Contrapositive: If 5 ≥ 3, then 7 ≥ 5. (True) Converse: If 5 < 3, then 7 < 5. (True)

Order of operations for compound propositions in absence of parentheses

1. ¬ (not) 2. ^ (and) 3. V (or)

If a compound proposition has n variables, then there are __ rows.

2^n rows example: 3 variables = 2^3 = 8 rows

Given a proposition: if 3 + 5 < 2, then 1 + 3 =/= 4 Write down the negation of the proposition.

3 + 5 >= and 1 + 3 = 4

rational number

A number r is rational if there exist integers x and y such that y≠0 and r=x/y.

Common word in proof: Thus and therefore

A statement that follows from the previous statement or previous few statements can be started with "Thus" or "Therefore".

Prove: There is an integer that can be written as the sum of the squares of two positive integers in two different ways

Let n=50. 50=12+72=52+52. Therefore, the integer 50 can be written as the sum of the squares of two positive integers in two different ways.■

Prove: Every positive integer is less than or equal to its square.

Let x be an integer, where x>0. Since x is an integer and x>0, then x≥1. Since x>0, we can multiply both sides of the inequality by x to get: x⋅x≥1⋅x Simplify the expression we get x^2 ≥x

Common word in proof: Let

New variable names are often introduced with the word "let". For example, "Let x be a positive integer".

Is 1 a prime number?

No

Is this a proposition? Take out the trash.

No

Is this a proposition? This soup is delicious.

No

Proposition? The math is hard.

No

Is the following logical expression a proposition: ∀z∃y⁢Q⁡(x,y,z)?

No, the variable x is not bound.

Predicate? 23 is a prime number.

No, this is a propositon, and it does not contain a variable.

∀x⁢P⁡(x) means??

P⁡(x) is true for every possible value of x in its domain

Prove: For every integer x, there is an integer y such that y+3=x.

Suppose that x is an integer. Let y=x−3. Since x is an integer, x−3 is also an integer. Therefore y is an integer. Furthermore, y+3=(x−3)+3=x. ■

negation

The opposite of a statement "¬"

Common word in proof: Suppose

The word "suppose" can also be used to introduce a new variable. For example: "Suppose that x is a positive integer". Suppose is also used to introduce a new assumption, as in: "Suppose that x is odd", assuming that x has already been introduced as an integer earlier in the proof.

Assume the propositions p, q, r have the following truth values: p is true q is true r is false What is the truth value for the following compound proposition? p V (not)q

True

Given a proposition: if 3 + 5 < 2, then 1 + 3 =/= 4 1. What is the truth value of the proposition?

True

Indicate whether each expression is true or false. 3∣12

True

Indicate whether the universally quantified statement is true or false. ∀x⁡(x^2>0)

True

r: If it is sunny out, I ride my bike. If it is sunny out, I ride my bike. It is not sunny out. I am not riding my bike. True or False?

True

In the following question, the domain is the set of all real numbers. x⁢y means x times y. ∃x∀y⁡(x⁢y=y)

True. If x is chosen to be 1, then x⁢y=y for any choice of y.

In the following question, the domain is the set of all nonnegative integers. x⁢y means x times y. ∃x∃y⁡(x⁢y=1)

True. When x=y=1, x⁢y=1.

Determine the truth value of the proposition. Then write down the negation of each proposition. ∃x∈R, x > x^2

True. ∀x∈R, x ≤ x^2

Determine the truth value of the proposition. Then write down the negation of each proposition. ∀x∈R, (x > 1) → (x^2 > x)

True. ∃x∈R, (x > 1) ^ (x^2 ≤ x)

Indicate whether each expression is true or false. 12 ∤ 3

True; There is no integer k such that 3=k·12, because 3/12 is not an integer. Therefore, 12 does not divide 3.

Determine the truth value of the proposition. Then write down the negation of the proposition. ∀x∈R, ∃y∈R, (x^2 + y^2) >= 0

True; ∃x∈R, ∀y∈R, (x^2 + y^2) < 0

disjunction

V "or"

Is 2 a prime number?

Yes

Is this a proposition? 4+8 = 36.

Yes

Is this a proposition? It will be sunny tomorrow.

Yes

Predicate? x is odd.

Yes

Proposition? 5 is an even number.

Yes

Proposition? The moon is made of cheese.

Yes

conjuction

^ "and"

axioms

a statement or proposition that is regarded as being established, accepted, or self-evidently true

free variable

a variable is free to take on any value in the domain

counterexample

an element in the domain for which the predicate is false

the proposition is always false, regardless of the truth value of the individual propositions that occur in it

contradiction

conditional operation

denoted with the symbol →. The proposition p → q is read "if p then q". The proposition p → q is false if p is true and q is false; otherwise, p → q is true.

If x divides y, then x is a __ of y

factor or divisor

prime number

if and only if n>1, and the only positive integers that divide n are 1 and n .

composite number

if and only if n>1, and there is an integer m such that 1<m<n and m divides n .

odd integer

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

even integer

if there is an integer k such that x = 2⁢k.

if a proposition the same truth value regardless of the truth values of their individual propositions

logically equivalent

Suppose that n is an integer. Is the number 2⁢n+3 even or odd?

odd

inclusive or

one or the other or both

p ^ q = ?

p and q

exclusive or

p and q evaluates to true when p is true and q is false or when q is true and p is false.

how to fill out truth table

p q r ------- T T T T T F T F T T F F F T T F T F F F T F F F

truth table for conditional operation

p | q | p -> q --------------- T T T T F F F T T F F T

truth table for biconditional operation

p | q | p ↔ q ---------------- T T T T F F F T F F F T

Converse

p → q is q → q

Inverse

p → q is ¬p → ¬q

Contrapositive

p → q is ¬q → ¬p

biconditional operation

p ↔ q "p if and only if q" must have same truth value

A statement with no free variables is a...

proposition

a statement that is either true or false

proposition

truth table for p and q

p|q| p ^ q ___________________ T T T T F F F T F F F F

truth table for p V q

p|q| pVq ___________________ T T T T F T F T T F F F

the proposition is always true, regardless of the truth value of the individual propositions that occur in it

tautology

bound variable

the variable is bound to a quantifier

a statement that can be proven to be true

theorem

∃x⁢P⁡(x) means what?

there exists an x, such that P(x); P⁡(x) is true for at least one possible value for x in its domain

parity

whether a number is odd or even

Select the statement that is equivalent to each statement given below. x >= is false

x < 3

Select the statement that is equivalent to each statement given below. x<−7 or x=−7

x <= -7

Suppose that r=0.677 and y=1000. Show that r is rational by giving an integer x such that r=xy.

x = 667

Select the statement that is equivalent to each statement given below. x < 3 is false.

x >= 3

x does not divide y

x ∤ y

x divides y

x|y

An integer x divides an integer y if and only if

x≠0 and y=k⁢x, for some integer k.

second De Morgan law

¬(p∧q)≡(¬p∨¬q) swaps the role of the disjunction and conjunction

first De Morgan law

¬(p∨q) ≡ (¬p ∧ ¬q)

De Morgan's law for quantified statements

¬∀x P(x) ≡ ∃x ¬P(x) ¬∃x P(x) ≡ ∀x ¬P(x)

De Morgan's law with nested quantifiers

¬∀x ∀y P(x, y) ≡ ∃x ∃y ¬P(x, y) ¬∀x ∃y P(x, y) ≡ ∃x ∀y ¬P(x, y) ¬∃x ∀y P(x, y) ≡ ∀x ∃y ¬P(x, y) ¬∃x ∃y P(x, y) ≡ ∀x ∀y ¬P(x, y)

Select the proposition that is logically equivalent to ∀x(¬P(x) ∧ Q(x)).

¬∃(P(x) ∨ ¬Q(x))

Is ∀x⁢P⁡(x) a proposition?

∀x⁢P⁡(x) is a proposition because it is either true or false

logically equivalent

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