Module 2: Logic and Reasoning

Question 1

What is a mathematical statement?

Answer 1

A mathematical statement is a statement that can be assigned a truth value and
classified as true or false, but not both.

Question 2

If all mathematical statements are declarative statements, are all declarative statements mathematical statements?

Answer 2

NO

Question 3

YES or NO the following statement is a mathematical statement: 7 is a lucky number.

Answer 3

NO

Question 4

YES or NO the following statement is a mathematical statement: Math 10 is a GE course.

Answer 4

YES

Question 5

The conjunction “p AND q” is written as _____

Answer 5

p ∧ q

Question 6

The disjunction “p OR q” is written as ____

Answer 6

p ∨ q

Question 7

The conditional “IF p THEN q” is written as _____

Answer 7

p → q

Question 8

In the conditional statement, p is referred to as the _____ and q as the ______

Answer 8

premise; conclusion

Question 9

The biconditional statement “p IF AND ONLY IF q” is written as _____

Answer 9

p ↔ q

Question 10

The negation statement “NOT p” is written as ____

Answer 10

~p

Question 11

Examples of delimiters to group statements together

Answer 11

(), {}, []

Question 12

When determining the truth value of statements with a delimeter, what statement do you evaluate first?

Answer 12

the statement inside the delimeters

Question 13

What is the truth value of the following statement, given that p,q, an r are true statements:
(~p ∧ q) ∨ ~(r → ~q)

Answer 13

true

Question 14

The conjunction p ∧q is true if both p and q are _____. Otherwise, it is false.

Answer 14

true

Question 15

The disjunction p ∨ q is true if ______ statement (p, q, or both) is true.
It is false only if ______.

Answer 15

at least one; both statements are false

Question 16

The conditional p → q is false only when the premise p is ____ and the
conclusion q is ____. Otherwise, it is true.

Answer 16

true; false

Question 17

The biconditional p ↔ q is true if p and q _______.

Answer 17

have the same truth value (both true or both false)

Question 18

The negation ~ p is true if p is _____. If p is ____, ~p is false.

Answer 18

false; true

Question 19

When constructing truth tables, what is the formula for computing the number of possible cases for all combinations of truth values?

Answer 19

let n be the number of statements; 2^n

Question 20

When are statements considered equivalent?

Answer 20

If they have the same truth value based on a truth table

Question 21

What is the negation of all?

Answer 21

not all, some, there is at least one

Question 22

What is the negation of “has more than”

Answer 22

has at most

Question 23

What is the negation of some?

Answer 23

all

Question 24

What is the negation of p ∨ q (p OR q)

Answer 24

not p AND not q

Question 25

Negate: We are winning the fight against poverty or everyone is in despair.

Answer 25

We are not winning the fight against poverty AND someone is not in despair.

Question 26

What is the negation of p ∧ q (p AND q)

Answer 26

not p OR not q

Question 27

Negate: The chairs are red and UP is at least 100 years old.

Answer 27

The chairs are not red or UP is less than 100 years old.

Question 28

Negate the following statement: ( p ∨ q ) ∧ (~r ∨ ~s)

Answer 28

(~p∧~q) ∨ (r∧s)

Question 29

What are the four forms or variants of the conditional statement?

Answer 29

original, inverse, converse, contrapositive

Question 30

Inverse form of the conditional statement

Answer 30

~p -> ~q

Question 31

Converse form of the conditional statement

Answer 31

q -> p

Question 32

Contrapositive form of the conditional statement

Answer 32

~q -> ~p

Question 33

Which variants of the conditional are equivalent?

Answer 33

inverse and converse; original and contrapositive

Question 34

What is the converse (p->q) statement: If it is raining, then I will bring my umbrella

Answer 34

If I bring my umbrella, then it is raining.

Question 35

What is the inverse (~p->~q) statement: If it is raining, then I will bring my umbrella

Answer 35

If it is not raining, then I will not bring my umbrella.

Question 36

What is the contrapositive (~q->~p) statement: If it is raining, then I will bring my umbrella

Answer 36

If I do not bring my umbrella, then it is not raining.

Question 37

Give 5 equivalent statements: If you care for the environment, then you should recycle.

Answer 37

1. (q if p): You should recycle if you care for the environment.
2. (p only if q): You care for the environment only if you recycle.
3. (q is necessary for p): Recycling is necessary for caring for the environment.
4. (all p are q): All who care for the environment should recycle.
5. (either not p or q): Either you do not care for the environment or you recycle.

Question 39

Valid argument forms

Answer 39

modus ponens, modus tollens, syllogism