Introduction to Proofs

Question 1

the proofs of theorems designed for human
consumption are almost always

Answer 1

informal proofs

Question 2

is a statement that can be shown to be true

Answer 2

theorem

Question 3

Less important theorems sometimes are called

Answer 3

propositions

Question 4

Theorems can also be referred to
as

Answer 4

facts or results

Question 5

We demonstrate that a theorem is true
with a

Answer 5

proof

Question 6

The statements used in a proof can include

Answer 6

axioms (or postulates)

Question 7

axioms (or postulates), which are

Answer 7

statements we assume to be true

Question 8

In practice, the
nal step of a proof is usually just the

Answer 8

conclusion of the theorem

Question 9

A less important theorem that is helpful in the proof of other results is called a

Answer 9

lemma
(plural lemmas or lemmata)

Question 10

is a theorem that can be established directly from a theorem that has been proved

Answer 10

corollary

Question 11

is a statement that is being proposed to be a true statement, usually on the basis of some partial
evidence, a heuristic argument, or the intuition of an expert

Answer 11

conjecture

Question 12

A ____________________ of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true.

Answer 12

direct proof

Question 13

The integer n is ______ if there exists an integer k such that n = 2k, and n is _____ if there exists an integer k such that n = 2k + 1. (Note that every integer is either even or odd, and no
integer is both even and odd.)

Answer 13

even, odd

Question 14

Two integers have the _____________ when both are even or both
are odd; they have ________________ when one is even and the other is odd.

Answer 14

same parity, opposite parity

Question 15

An integer a is a _________________ if there is an integer b such that a = b2

Answer 15

perfect square

Question 16

Proofs of theorems of this type that are not direct
proofs, that is, that do not start with the premises and end with the conclusion, are called

Answer 16

indirect proofs.

Question 17

An extremely useful type of indirect proof is known as

Answer 17

proof by contraposition

Question 18

if we can show that p is false, then we have a proof, called a

Answer 18

vacuous proof, of the conditional statement p → q

Question 19

We can also quickly prove a conditional statement p → q if we know that the conclusion

Answer 19

q is true

Question 20

A proof of p → q that uses the fact that q is true is called a

Answer 20

trivial proof

Question 21

The real number r is _________if there exist integers p and q with q = 0 such that r = p/q. / suppose to be through first = sign

Answer 21

rational

Question 22

A real number that is not rational is called

Answer 22

irrational

Question 23

Because the statement r ∧ ¬r is a contradiction whenever r is a proposition, we can prove that p is true if we can show that ¬p → (r ∧ ¬r) is true for some proposition r. Proofs of this
type are called

Answer 23

proofs by contradiction

Question 24

a statement of the form ∀xP (x) is false, we need only find a

Answer 24

counterexample

Question 25

Every positive integer is the sum of the

Answer 25

squares of two integers”

Question 26

Many incorrect arguments are based on a fallacy called

Answer 26

begging the question

Question 27

begging the question fallacy is also called

Answer 27

circular reasoning

Question 28

circular reasoning.

Answer 28

fallacy arises when a statement is proved using itself, or a statement equivalent to it