proof

Question 1

notations to use in proof and the rest of maths a-level

Answer 1

• a set is a collection (shown in the curly brackets {} )
• arrow symbols are logic symbols (means one thing implies the other)
• there is a variety of equal signs

Question 2

A set can be written in different ways…

Answer 2

• a list of elements
• a rule e.g odd numbers between 0 and 10
• mathematical notations e.g {x:x<0} i.e the set of values of x such that x is less than 0.

Question 3

3 ways of writing the equal sign

Answer 3

• the wiggly one means approximately
• one with a line through means does not equal
• three lines means identically equal to.

Question 4

proof by exhaustion

Answer 4

• means you break it down into two cases
• cover all solutions
• prove separate that the statement is true
• example it asks that the statement is an always an odd number, you need to split that into odd and even

Question 5

proof by deduction

Answer 5

when you use known facts to build up you argument and show a statement must be true.

e.g “the product of two rational numbers is always a rational number”
- Use the definition that a rational number that can be written as a quotient of two integers, where the denominator is non-zero.

Question 6

Disproof by counter-example

Answer 6

• easiest way to show that the mathematical statement is false
• all you do is find a statement where the statement doesn’t hold.

Question 7

proof by contradiction

Answer 7

• you say ‘Assume that the statement is not true’ then prove that something is impossible would have to be true for that case.

e.g prove: ‘if x^2 is even then x must be even’
- say, Assume this is not true. There must be an odd number for x where x^2 is even

Question 8

surds are irrational

Answer 8

• you can use proof by contradiction to prove some really important facts
• e.g you can prove that the square root of any non-square number is irrational

Question 9

proving infinitely many of something

Answer 9

• you can use proof by contradiction to show that there are infinitely many numbers in a certain set
  e.g even, odd, multiples etc.

say you are proving there are infinitely many even numbers
- say that they are finite and that N is the biggest possible N value
where N=2n (n is an integer)
- if u add 2 you get, N+2 = 2n+2 = 2(n=1)
- which is then bigger then N and therefore you have contradicted your initial assumption