symbol "∃" name and meaning [notation]

"∃" is called the existential quantifier;

means "there exists"

or "there exist".

symbol "∀" name and meaning [notation]

"∀" is called the universal quantifier;

means "for all"

or "for each"

or "for every"

or "whenever" .

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 .

symbol "∄" meaning [notation]

"∄" means "there does not exist".

symbol "≡" meaning [notation]

"≡" denotes logical equivalence.

symbol "∃!" meaning [notation]

"∃!" means "there exists a unique".

two statements that are equivalent to a statement of the form

(∃!*n* ∈ **N** such that)

[notation]

(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)

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* .

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

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*”)

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__ .

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__ .

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__ .

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__ .

symbol "⇔" name and meaning [notation]

"⇔" is called the double implication symbol;

means "if and only if".

four statements equivalent to "*P* ⇔ *Q*" [notation]

"*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*)) .

converse of P ⇒ Q [notation]

The converse of P ⇒ Q

is "Q ⇒ P" .

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.

contrapositive of P ⇒ Q [notation]

The contrapositive of P ⇒ Q

is (not P) ⇒ (not Q) .

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

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

[notation applicaton]

True

negation of the statement "*P*"

The negation of "*P*" is "not *P*".

" ¬*P* " meaning [notation]

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

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

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

De Morgan's laws [notes]

De Morgan's laws:

¬(*P* __or__ *Q*) ≡ (¬*P* __and__ ¬*Q*)

¬(*P* __and__ *Q*) ≡ (¬*P* __or__ ¬*Q*)

"P ⇒ Q" in negation notation [notation]

P ⇒ Q

is equivalent to

¬P or Q .

negation of P ⇒ Q [notation application]

The negation of

¬(P ⇒ Q)

is

(P and ¬Q) .

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).