Chapter 0.2: Mathematical Statements

Question 1

What is a statement?

Answer 1

Any declarative sentence which is either true or false

Question 2

What are the two types of statements?

Answer 2

Atomic; cannot be divided not smaller statements. Molecular.

Question 3

Molecular statements can be built by combining atomic statements using ____.

Answer 3

Logical connectives

Question 4

What are binary connectives? List them?

Answer 4

Connectives that connect two statements: conjunction, disjunction, implication, biconditional

Question 5

What are unary connectives? List them?

Answer 5

Applies to a single statement only: negation.

Question 6

When trying to figure out the ____ of a molecular statement we only need to know the truth values of its atomic components combined with the logical connective

Answer 6

Truth value

Question 7

List the 4 propositional variables

Answer 7

P Q R S

Question 8

List the 5 logical connectives and their plain English counterparts and their Symbol

Answer 8

Conjunction; “And”; ∧

Disjunction; “Or”; ∨

Implication; “if…, then…; –>

Biconditional; “if and only if”;

Negation; “Not”; ¬

Question 9

What does it mean that the disjunction “or” is inclusive?

Answer 9

Both statements can be true as well

Question 10

When is a conjunction true?

Answer 10

When both statements are true.

Question 11

When is a disjunction true?

Answer 11

When either, or both statements are true

Question 12

When is an implication true?

Answer 12

When the P is false, or if Q is true or both.

Question 13

When is a biconditional true?

Answer 13

When both statements have the same truth value

Question 14

When is a negation true?

Answer 14

When the statement is false.

Question 15

In an implication, what are P and Q called? Include alternatives.

Answer 15

P: hypothesis or antecedent

Q: conclusion or consequent

Question 16

What is the converse of the implication P –> Q?

Answer 16

Q –> P

Question 17

T/F: The truth value of the converse is dependent on the truth value of the original implication.

Answer 17

F

Question 18

What is the contrapositive of P –> Q?

Answer 18

¬Q –> ¬P

Question 19

When P –> Q and Q –> P are both true, we can say that Q and P are ____.

Answer 19

Biconditionals

Question 20

“If and only if” statements must have a forward direction and a backwards direction that are both ___.

Answer 20

True

Question 21

What does “P is necessary for Q” translate into symbolically?

Answer 21

Q –> P

Question 22

What does “P is sufficient for Q” translate into symbolically?

Answer 22

P –> Q

Question 23

What does P is sufficient and necessary for Q” translate into symbolically?

Answer 23

P Q

Question 24

If you have an assumption, you must think about …

Answer 24

What is necessary to make that hypothesis true

Question 25

What is this P(n) --\> ¬P(n + 7)? What makes it so?

Answer 25

A predicate, it has free variables.

Question 26

What are free variables?

Answer 26

Variables that have yet to be specified

Question 27

What is the symbol and plain English translation for an existential quantifier?

Answer 27

∃, reads "there exists" or "there is."

Question 28

What is the symbol and plain English translation for a universal quantifier?

Answer 28

∀, reads "for all" or "every"

Question 29

Convert this into plain English: ∀x∃y(y \< x). what is this an example of?

Answer 29

“For all x there exists y in which y is less than x.” This is a quantified statement

Question 30

What is the Domain of discourse for a quantified statement?

Answer 30

The values being considered when we say “all”

Question 31

What is the default Domain of discourse in discrete math?

Answer 31

All-natural numbers {0, 1, 2, …}

Question 32

¬∀xP(x) is equivalent to \_\_\_\_\_. Say both quantified statements in plain English.

Answer 32

¬∀xP(x) is equivalent to **∃x¬P(x)** "If not everything has a property, then something doesn’t have a property."

Question 33

¬∃xP(x) is equivalent to \_\_\_\_\_. Say both quantified statements in plain English.

Answer 33

¬∃xP(x) is equivalent to **∀x¬P(x)** "If there is not something with a property, then everything doesn’t have that property"

Question 34

Assume that sentences containing predicates with free variables are intended as statements, where…

Answer 34

the variables are universally quantified.

Question 35

\_\_\_\_ is a statement that cannot be true or false?

Answer 35

Imperative statement