Proofs

Question 1

How do you proof a triangle is a right angle triangle using coordinates?

Answer 1

1. Find the gradient of each line of the triangle using the coordinates
2. See if the two of the gradients are negative reciprocal
3. This means they are perpendicular and therefore right angle to each other

Question 2

prove that n²-n is an even number for all values of n

Answer 2

n² − n = n(n − 1)
1. If n is even, n − 1 is odd
and even × odd = even

2. If n is odd, n − 1 is even
   and odd × even = even

∴ n² − n is even for all values of n.

Question 3

How can you show that an equation (3n² − 4n + 10 for example) is positive/negative/equal to a value for all values?

Answer 3

1. complete the square
2. number outside the bracket is the minimum number for all values which is positive/negative/equal to a value
3. ∴ positive/negative/equal to a value for all values

Question 4

why would k be 0≤ in x² + 8x + 20?

Answer 4

When k = 0:
(0)x2 + 5(0)x + 3 = 0
3 = 0
which is impossible
So combining these:
0 ≤ k…

Question 5

What are the types of mathmatical proofing?

Answer 5

• Consequence and Equivalence
• Proofing by Exhaustion
• Proofing by Deduction
• Proofing by Contradiction
• Disproofing by Counter Example

Question 6

How do you proof using Consequence and Equivalence?

Answer 6

Consequence:
* you prove that one statement implies the other (a=b, not b=a)
* represented by ⇒
Equivalence:
* you prove that both statements imply each other (a=b and b=a)
* represented by ⇐⇒

Question 7

How do you proof using Proof by Exhaustion?

Answer 7

use different examples (for example numbers) to prove a statemtent

Question 8

What is a conjective?

Answer 8

a statement that you believe is correct, but needs to be proved

Question 9

How can every integer be represented algebraically?

Answer 9

n=3k
n=3k-1
n=3k-2

Question 10

How do you proof using Proof by Deduction?

Answer 10

you use logical statements that would support the conjective if correct, in order to prove it

Question 11

How do you proof using Disproof by Counter Example?

Answer 11

• show that a conjective is incorrect using an example
• by subing in numbers for example

Question 12

How do you prove that for every real value of x a quadratic is > 0

Answer 12

find the minimum point and show that it’s above (0,0) or positive