Chapter 3

Question 1

symbol “∃” name and meaning [notation]

Answer 1

“∃” is called the existential quantifier;
means “there exists”
or “there exist”.

Question 2

symbol “∀” name and meaning [notation]

Answer 2

“∀” is called the universal quantifier;
means “for all”
or “for each”
or “for every”
or “whenever” .

Question 3

two components of a quantified statement [notation]

Answer 3

(1) quanitfied segments (ex: “(∃0 ∈ Z such that)(∀a ∈ Z)”)
(2) final statement (ex: “a + 0 = a”)

together, make the quantified statement…

(∃0 ∈ Z such that)(∀a ∈ Z) a + 0 = a .

Question 4

symbol “∄” meaning [notation]

Answer 4

“∄” means “there does not exist”.

Question 5

symbol “≡” meaning [notation]

Answer 5

“≡” denotes logical equivalence.

Question 6

symbol “∃!” meaning [notation]

Answer 6

“∃!” means “there exists a unique”.

Question 7

two statements that are equivalent to a statement of the form
(∃!n ∈ N such that)
[notation]

Answer 7

(1) existence statement (ex: “(∃n ∈ N such that)”)
(2) uniqueness statement (ex: “(if n ∈ N and m ∈ N both have the given property, then n = m)”)

These two statements together mean the same as the uniqueness statement of the form

(∃!n ∈ N such that)

Question 8

three statements equivalent to “P ⇒ Q” [notation]

Answer 8

“P ⇒ Q” can also be written…

(i) P implies Q .
(ii) If P, then Q .
(iii) (not P) or Q .

Question 9

If P is a false statement, then the implication P ⇒ Q is … (true/false) [notation]

Answer 9

If P is a false statement, then the implication P ⇒ Q is true.

(follows from the fact that “P ⇒ Q” is equivalent to “(not P) or Q”)

Question 10

P is true;

Q is true;

then P ⇒ Q is … (true/false) .

[truth table]

Answer 10

P is true;

Q is true;

then P ⇒ Q is true .

Question 11

P is true;

Q is false;

then P ⇒ Q is … (true/false) .

[truth table]

Answer 11

P is true;

Q is false;

then P ⇒ Q is false .

Question 12

P is false;

Q is true;

then P ⇒ Q is … (true/false) .

[truth table]

Answer 12

P is false;

Q is true;

then P ⇒ Q is true .

Question 13

P is false;

Q is false;

then P ⇒ Q is … (true/false) .

[truth table]

Answer 13

P is false;

Q is false;

then P ⇒ Q is true .

Question 14

symbol “⇔” name and meaning [notation]

Answer 14

“⇔” is called the double implication symbol;
means “if and only if”.

Question 15

four statements equivalent to “P ⇔ Q” [notation]

Answer 15

“P ⇔ Q” can also be written…

(i) P if and only if Q .
(ii) (P ⇒ Q) and (Q ⇒ P) .
(iii) Either P and Q are both false, or P and Q are both true.
(iv) (P and Q) or ((not P) and (not Q)) .

Question 16

converse of P ⇒ Q [notation]

Answer 16

The converse of P ⇒ Q

is “Q ⇒ P” .

Question 17

True or false: an implication and its converse are equivalent.

(If false, correct the statement to make it true.)

[notation application]

Answer 17

False

Correction: An implication and its converse are not generally equivalent.

Question 18

contrapositive of P ⇒ Q [notation]

Answer 18

The contrapositive of P ⇒ Q

is (not P) ⇒ (not Q) .

Question 19

True or false: an implication and its contrapositive are equivalent.

(If false, correct the statement to make it true.)

[notation applicaton]

Answer 19

True

Question 20

negation of the statement “P”

Answer 20

The negation of “P” is “not P”.

Question 21

” ¬P “ meaning [notation]

Answer 21

” ¬P “ means “not P” (ie negation of statement P).

Question 22

negation of “and” and “or” [notes]

Answer 22

Negation interchanges “and” and “or” according to De Morgan’s laws.

Question 23

De Morgan’s laws [notes]

Answer 23

De Morgan’s laws:

¬(P or Q) ≡ (¬P and ¬Q)

¬(P and Q) ≡ (¬P or ¬Q)

Question 24

“P ⇒ Q” in negation notation [notation]

Answer 24

P ⇒ Q

is equivalent to

¬P or Q .

Question 25

negation of P ⇒ Q [notation application]

Answer 25

The negation of

¬(P ⇒ Q)

is

(P and ¬Q) .

Question 26

algorithm for negating a quanifying statement [steps]

Answer 26

To negate a quantifying statement…

(1) Maintain the order of the quantified segments.
(2) Change every (∀…) segment into a (∃… such that) segment.
(3) Change each (∃… such that) segment into a (∀…) segment.
(4) Negate the final statement (ex: change = to ≠ , > to < , etc).