Proofs

Question 1

theorem

Answer 1

statement that can be proven to be true

Question 2

proof

Answer 2

a series of steps, each of which follows logically from assumptions

Question 3

axioms

Answer 3

statements assumed to be true

Question 4

proof by exhaustion

Answer 4

prove the statement by checking each element individually

Question 5

counterexample (chapter 4)

Answer 5

an assignment of values to variables that shows that a universal statement is false

Question 6

direct proof

Answer 6

hypothesis p is assumed to be true and the conclusion c is proven as a direct result of the assumption

Question 7

proof by contrapositive

Answer 7

proves conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true

Question 8

even integer

Answer 8

2k

Question 9

odd integer

Answer 9

2k+1

Question 10

proof by contradiction

Answer 10

assume the theorem is false and then show that some logical inconsistency arises as a result of this assumption, also known as indirect proof

Question 11

proof by cases

Answer 11

∀x P(x) breaks the domain for the variable x into different classes and gives a different proof for each class

Question 12

set identity

Answer 12

an equation involving sets that is true regardless of the contents of the sets in the expression