2.1 ) Proof

Question 1

What are the different number sets ?

Answer 1

• Natural numbers ( N )
• Whole numbers ( W )
• Integers ( Z )
• Rational numbers ( Q )
• Irrational numbers ( R / Q )
• Real numbers ( R )

Question 2

What are Natural numbers ( N ) ?

Answer 2

All positive integers not including zero

Question 3

What are whole numbers ( W ) ?

Answer 3

All positive integers including zero

Question 4

What are Integers ( Z ) ?

Answer 4

All positive and negative integers, including zero

Question 5

What are rational numbers ( Q )?

Answer 5

• All positive/ negative integers including zero
• Recurring/ terminating decimals and fractions

Question 6

What are irrational numbers ( Q / R ) ?

Answer 6

Surds, pi, e …

Question 7

What are real numbers ( R ) ?

Answer 7

All numbers apart from complex / imaginary numbers

Question 8

What are the different methods for proof ?

Answer 8

• Proof by counter example
• Proof by deduction
• Proof by exhaustion
• Proof by contradiction

Question 9

How do you write odd and even integers ?

Answer 9

• EVEN : 2k
• ODD : 2m + 1

Question 10

How do you write consecutive integers ?

Answer 10

n, n + 1, n + 2, etc…

Question 11

What are coprime integers ?

Answer 11

• Two integers whose highest common factor is 1
• Eg. 18 and 25
• Not 12 and 18 since they both share 6

Question 12

Proof by deduction that the sum of three consecutive integers is always a multiple of three ?

Answer 12

Question 12

What is a composite integer ?

Answer 12

• A positive integer greater than 1 that has at least one factor other than 1 and itself
• Smallest composite integer is 4 ( divisible by 1,2 and 4 )

Question 13

Prove by exhaustion that no square number ends in 2, 3, 7 or 8 ?

Answer 13

Question 14

Disproof by counter-example that the expression n^2 + n + 5 is prime for all values of n ?

Answer 14

Question 15

Proof by contradiction that /2 is irrational ?

Answer 15

Question 16

Proof by contradiction that logv3^2 is irrational ?

Answer 16

Question 17

Prove there are infinitely many even numbers ?

Answer 17

Question 18

Answer 18

Question 19

What symbol should be used in the second line ?

Answer 19

• x = 0 and x = 3 are both solutions
• Therefore the symbol should be <=, since it shows x =3 is a solution but not the only solution