Proof

Question 1

5 main types of proofs

Answer 1

Question 2

what must you include in a prove or show that question

Answer 2

Question 3

what is a conjecture

Answer 3

Question 4

what is a mathematical implication

Answer 4

Question 5

how can an even and a odd number be expressed

Answer 5

Question 6

what are the ways to work with consecutive integers

Question 7

how can rational numbers be written

Answer 7

Question 8

3 ways to prove LHS=RHS

Answer 8

Question 9

proof by exhaustion example

Answer 9

steps:
1) make n = to another non multiple of 3
2) expand and prove it is not divisible by 3
3) repeat
4) conclude using the original statement

Question 10

proof by exhaustion definition

Answer 10

Question 11

dispoof by counter example definition

Answer 11

Question 12

Disproof by counter example, example

Answer 12

steps:
1) guess a positive whole number and set it = n
2) expand
3) conclude

it is a guessing game since for a conjecture to be true it must be true in all cases

Question 13

proof by deduction definition

Answer 13

Question 13

tips (steps) for how to solve proof by deduction

Answer 13

Question 14

proof by deduction example

Answer 14

Question 15

proof by contradiction example 1

Answer 15

steps:
1) set the irrational number = a/b
2) state that a/b is coprime (simplest as possible cannot be divided anymore)
3) square both sides and solve for a
4) state that a is a multiple of the irrational number
5) substitute a in as irrational number X p
6) solve for b and state that it is also a multiple of the irrational number
7) conclude that if they are both a multiple of the irrational number they are not coprime and therefore the irrational number is irrational

Question 16

prove by contradiction example 2

Answer 16

1) contradict the initial statement
2) list the finite number of primes
3) set n = to the list
4) state that n is not divisible by any of the list (always has a remainder of 1)
5) state that n is larger than Pn and n is a prime not in our finite list

Question 17

prove by contradiction, example 3

Answer 17

steps:
1) assume the opposite of the statement
2) already know a, so find r
3) once finding what r = substitute into the next set of values
4) expand and make an equation
5) show that there are no solutions using the discriminant < 0
6) conclude that there are no real values and the initial statement is true

Question 18

proof by contradiction, example 4

Answer 18

steps:
1) assume the opposite of the statement
2) differentiate and solve to find stationary points
3) show that sinx must be between -1 and 1 due to the graph
4) but what it is equal to is greater than 1
5) therefore conclude that the curve has no stationary points