definitions

Question 1

a divides b

Answer 1

there is a natural number k such that b = ak

Question 2

a natural number p is prime if…

Answer 2

p is greater than 1 and the only numbers that divide p are 1 and p

Question 3

a natural number that is COMPOSITE is…

Answer 3

neither 1 nor prime

Question 4

For f(a) = b, the image is…

Answer 4

b

Question 5

For f(a) = b, the pre-image is…

Answer 5

a

Question 6

a proposition is

Answer 6

a sentence that has exactly one truth value, either T or F

Question 7

the negation of a proposition P

Answer 7

is the proposition “not P”, which is true when P is false

Question 8

given propositions P and Q, the conjunction of P and Q is true when…

Answer 8

both P and Q are true

Question 9

given propositions P and Q, the disjunction of and Q is true when…

Answer 9

exactly at least one of P or Q is true

Question 10

a tautology is…

Answer 10

a propositional form that is true for every assignment of truth values to its components
(ex. P∨~P)

Question 11

a contradiction is…

Answer 11

a propositional form that is false for every assignment of truth values to its components
(ex. P∧~P)

Question 12

two propositional forms are equivalent if…

Answer 12

they have the same truth tables

Question 13

a denial of any proposition P is

Answer 13

any proposition equivalent to ~P (ex. ~~~P)

Question 14

the conditional sentence, “if P, then Q” is true if…

Answer 14

P is false or Q is true

Question 15

the conditional sentence, “if P then Q” is false if…

Answer 15

P is true and Q is false

Question 16

the converse of P⇒Q is

Answer 16

Q⇒P

Question 17

the contrapositive of P⇒Q

Answer 17

~Q ⇒ ~P

Question 18

T or F: a statement and its converse are always equivalent

Answer 18

F

Question 19

T or F: a statement and its contrapositive are equivalent

Answer 19

T

Question 20

for propositions P and Q, the biconditional statement is…

Answer 20

the statement “P if and only if Q”

Question 21

a biconditional sentence is true when

Answer 21

P and Q have the same truth values

Question 22

open sentence

Answer 22

a sentence that contains variables, becomes a proposition when its variables are assigned specific values

Question 23

truth set

Answer 23

the collection of objects that may be substituted to make an open sentence a true proposition

Question 24

universe of discourse

Answer 24

the collection of objects that are available for consideration, often number systems ℂ, ℕ, ℚ, ℝ, ℤ

Question 25

two open sentences are equivalent when…

Answer 25

they have the same truth set

Question 26

T or F: the sentence (∃x)P(x) is always true

Answer 26

F, only true if P(x) is nonempty

Question 27

T or F: (∀x)P(x) is always true

Answer 27

F, only true if the truth set of P(x) is the entire universe

Question 28

two quantified statement are equivalent in a given universe if

Answer 28

they have the same truth value in that universe

Question 29

two quantified statements are equivalent if

Answer 29

they are equivalent in every universe

Question 30

(∃!x) P(x) is true when

Answer 30

the truth set of P(x) has exactly one element

Question 31

theorem

Answer 31

a statement that describes a pattern or relationship among quantities or structures

Question 32

a proof of a theorem is

Answer 32

a justification of the truth of the theorem that follows the principles of logic

Question 33

axioms (or postulates)

Answer 33

a set of statement that are assumed to be true

Question 34

in any proof at any time you may…

Answer 34

state an axiom, an assumption, or a previously proven result;
use the tautology rule;
use the replacement rule;
use a definition to state an equivalent to a statement earlier in the proof;
modus ponens rule

Question 35

tautology rule

Answer 35

can state a sentence whose symbolic translation is a tautology

Question 36

replacement rule

Answer 36

state a sentence equivalent to any statement earlier in the proof

Question 37

lemma

Answer 37

a result that serves as a preliminary step, “stepping stone” to final result

Question 38

modus ponens rule

Answer 38

after statements P and P⇒Q appear, state Q

Question 39

direct proof of P⇒Q

Answer 39

assume P… therefore Q
Thus, P⇒Q

Question 40

proof of P⇒Q by contrapositive

Answer 40

assume ~Q…
therefore ~P
thus ~Q⇒~P
therefore P⇒Q

Question 41

when would you use proof by CP?

Answer 41

statements of either p and Q is negation or the connection berween denials is easier to understand

Question 42

proof of P by contradiction

Answer 42

suppose ~P…
therefore Q…
therefore ~Q.
hence Q∧~Q is a contradiction
thus P

Question 43

when might you use to do a proof by contradiction?

Answer 43

to prove any proposition P since direct proofs and proofs by CP can only be used for conditionals, proof by contradiction can also be use for conditionals

Question 44

direct proof of (∀x) P(x)

Answer 44

let x be an arbitrary object in the universe…
hence P(x) is true.
therefore, (∀x) P(x) is true

Question 45

proof of (∀x) P(x) by contradiction

Answer 45

suppose ~(∀x) P(x)
then (∃x)~P(x)
let t be an object such that ~P(t)…
therefore Q∧~Q which is a contradiction
thus (∃x)~P(x) is false so (∀x) P(x) is true

Question 46

constructive proof of (∃x)P(x)

Answer 46

specify one particular object a
if necessary verify that a is in the universe…
therefore, P(a) is true.
thus, (∃x)P(x)

Question 47

indirect proof of (∃x)P(x)

Answer 47

… therefore there must be an object a such that P(a) is true.
therefore (∃x)P(x) is true

Question 48

proof of (∃x)P(x) by contradiction

Answer 48

suppose ~(∃x)P(x)
then (∀x) ~P(x)…
therefore ~Q∧Q which is a contrdiction
thus ~(∃x)P(x) is false
therefore, (∃x)P(x) is true