Chapter 3 Flashcards Preview

MAT 1362 > Chapter 3 > Flashcards

Flashcards in Chapter 3 Deck (26)
Loading flashcards...
1

symbol "∃" name and meaning [notation]

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

2

symbol "∀" name and meaning [notation]

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

3

two components of a quantified statement [notation]

(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 .

 

 

4

symbol "∄" meaning [notation]

"∄" means "there does not exist".

5

symbol "≡" meaning [notation]

"≡" denotes logical equivalence.

6

symbol "∃!" meaning [notation]

"∃!" means "there exists a unique".

7

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

(1) existence statement (ex: "(∃n ∈ N such that)")

(2) uniqueness statement (ex: "(if nN and mN both have the given property, then n = m)")

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

(∃!nN such that)

8

three statements equivalent to "P ⇒ Q" [notation]

"P ⇒ Q" can also be written...

(i) P implies Q .

(ii) If P, then Q .

(iii) (not P) or Q

9

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

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

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

10

P is true;

Q is true;

then P ⇒ Q is ... (true/false) .

[truth table]

P is true;

Q is true;

then P ⇒ Q is true .

11

P is true;

Q is false;

then P ⇒ Q is ... (true/false) .

[truth table]

P is true;

Q is false;

then P ⇒ Q is false . 

12

P is false;

Q is true;

then P ⇒ Q is ... (true/false) .

[truth table]

P is false;

Q is true;

then P ⇒ Q is true .

13

P is false;

Q is false;

then P ⇒ Q is ... (true/false) .

[truth table]

P is false;

Q is false;

then P ⇒ Q is true .

14

symbol "⇔" name and meaning [notation]

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

15

four statements equivalent to "PQ" [notation]

"PQ" 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)) .

 

16

converse of P ⇒ Q [notation]

The converse of P ⇒ Q

is "Q ⇒ P" .

17

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

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

[notation application]

False

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

18

contrapositive of P ⇒ Q [notation]

The contrapositive of P ⇒ Q

is (not P) ⇒ (not Q) . 

19

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

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

[notation applicaton]

True

20

negation of the statement "P"

The negation of "P" is "not P".

21

" ¬P " meaning [notation]

" ¬P " means "not P" (ie negation of statement P).

22

negation of "and" and "or" [notes]

Negation interchanges "and" and "or" according to De Morgan's laws.

23

De Morgan's laws [notes]

De Morgan's laws:

¬(P or Q) ≡ (¬P and ¬Q)

¬(P and Q) ≡ (¬P or ¬Q)

 

24

"P ⇒ Q" in negation notation [notation]

P ⇒ Q

is equivalent to

¬P or Q . 

25

negation of P ⇒ Q [notation application]

The negation of

¬(P ⇒ Q)

is

(P and ¬Q) .

 

26

algorithm for negating a quanifying statement [steps]

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).