Numbers and Proof

Question 1

Proof By Induction

Aim

Answer 1

to show an assertion is true for all integers n>=n0 where n0 is usually 0 or 1

Question 2

Proof By Induction

Method

Answer 2

1) Prove the assertion for n0
2) The Induction Assumption, assume the assertion is valid for some number n>=n0
3) Prove using the induction assumption that the assertion is true valid for the number n+1

Question 3

Proof by Contradiction / Indirect Proof

Method

Answer 3

1) Assume that there exists a contradiction to the claim you are trying to prove
2) start reasoning from this assumption until you reach a contradiction, i.e. something that is impossible based on your other assumptions

Question 4

Rational Numbers

Answer 4

Q = {r=p/q, p,q ε Z}

-it is usually assumed that these two integers are coprime, i.e. they share no common factors

Question 5

Irrational Numbers

Answer 5

An irrational number rεR can be viewed as a limit of a sequence of rational numbers

Question 6

Real Numbers

Answer 6

Real numbers are pictured via the real line as a continuous spectrum with no gaps

Question 7

Natural Numbers

Answer 7

N = {1, 2, 3, ….}

Question 8

Integers

Answer 8

Z = {…, -2, -1, 0, 1, 2, …}

Question 9

Union

Answer 9

∪, the union of two sets is a new set that contains all of the elements that are in at least one of the two sets

Question 10

Intersection

Answer 10

∩, the intersection of two sets is a new set that contains all of the elements that are in both of the two sets