Language of Proof

Question 1

2 synonyms for a STATEMENT

Answer 1

proposition, premise

Question 2

4 types of statements

Answer 2

(1) simple sentence
(2) mathematical declaration
(3) assertion/definition
(4) deduction/claim to be proven

Question 3

a statement’s logical value is…

Answer 3

either TRUE or FALSE but not both

Question 4

a proven statement is…

Answer 4

a statement shown to be true

Question 5

proof is…

Answer 5

the evidence used to establish a proven statement

Question 6

a counterexample is…

Answer 6

an example that disproves (falsifies) a statement

Question 7

what does negation do

Answer 7

changes a statement’s logical value

Question 8

negation of P is…

Answer 8

“not P”

¬P or ~P or P’ or P̄

Question 9

the negation of P and Q

Answer 9

not P nor Q
¬(P and Q) = ¬P or ¬Q

Question 10

the negation of P or Q

Answer 10

not P and not Q
¬(P or Q) = ¬P and ¬Q

Question 11

the negation of P⟹Q

Answer 11

P and ~Q
there exists a case where P and not Q

(the condition that causes P to imply Q no longer exists)

Question 12

an implication is…

Answer 12

an IF-THEN statement;
a conditional statement

Question 13

the notation for “if P then Q”

Answer 13

P⟹Q
“P implies Q”

Question 14

2 other ways to say “if P then Q”

Answer 14

(1) P is a SUFFICIENT condition to conclude Q
(2) Q is NECESSARY if P

Question 15

what does a converse statement do

Answer 15

reverses the implication

Question 16

the converse of P⟹Q

Answer 16

Q⟹P

Question 17

two statements are EQUIVALENT (aka are an equivalence) when…

Answer 17

the statement and its converse are both true

Question 18

notation for equivalent statements

Answer 18

P⇔Q
P if and only if Q
P iff Q
(and all vice versa)

Question 19

what a contrapositive does

Answer 19

(1) negates and (2) swaps both parts of a IF-THEN statement

Question 20

the contrapositive of P⟹Q (if P then Q)

Answer 20

¬Q⟹¬P
(if not Q then not P)

Question 21

why original and contrapositive statements are an equivalence

Answer 21

if P⟹Q is true, then ¬Q⟹¬P is also true (kind of like two negatives makes one positive)

e.g. If I’m 80 years old then I’ll die ⇔ if I’m not dead then I’m not 80 years old yet

Question 22

abbreviation for “if and only if”

Answer 22

iff

Question 23

what do quantifiers do

Answer 23

quantify (put a number on) how many values a variable can take on from a specified set

e.g. x is rational for all x∈R

Question 24

universal quantifier:
(1) meaning
(2) notation

Answer 24

(1) “for all” x
(2) ∀x

Question 25

existential quantifier
(1) meaning
(2) notation

Answer 25

(1) “there exists” x that…
(2) ∃x

Question 26

abbreviation for “such that”

Answer 26

s.t. or :

Question 27

RTP means

Answer 27

required to prove

Question 28

QED means

Answer 28

demonstrated as required (quod erat demonstrandum)

Question 29

~∀ =

Answer 29

∃

Question 30

~∃ =

Answer 30

∀

Question 31

translate into words:
∀n∈Z, ∃m∈Z, s.t. (n+m)/5 ∈Z

Answer 31

for all integers n, there exists integer m, such that (n+m) divided by 5 is an integer—aka n+m is a multiple of 5

Question 32

translate into words:
∀x∈A, ∃y∈B s.t. P(x,y)

Answer 32

for all x in set A, there exists y in set B such that (statement P)

Question 33

~[∀x∈A, ∃y∈B s.t. P(x,y)] ⇔

Answer 33

∃x∈A, ∀y∈B s.t. ~P(x,y)

Question 34

translate into words:
∃x∈A, ∀y∈B s.t. ~P(x,y)

Answer 34

there exists x in set A for all y in set B, such that (statement P) is not the case