Types of proof

1. Proof by Construction (building)
To proof a theorem use to demonstrate how that object are construction or building.

This method is called proof by construction

2. Proof by Contradiction (disagreement)
We assume that the statement is false then show that assumption leads to an obviously false consequences

Example: Prove √2= is irrational
• Let’s suppose √2 were a rational number
• So, √2 =m/n, where m and n not divisible by any number then 1.
• We can say that both of m and n can’t be even
• 2=m2/n2 => m2=2.n2 for this , we can say m is even
As m is even, we can say m=2k
• 2n2=m2
Form (1) => (2k)2 = 2n2
=> 4k2 = 2n2
=> n2 = 2k2
So, n also even

• But m and n both can’t be even according to our assumption
• We are conclude our assumption false are conclude our assumption is false √2 can’t be re irrational.

3. Proof by Induction (introduction)
These types of proof consist of two parts induction steep and basis.

At, first we have to proof for p(1) statement is true [basis] then by considering p(m) is true we have to proof , for p(m+1) statement is also true.

Example:
1+2+3+ ………………… +n =n (n+1)/2

Basic:
For p (1)
LHS = 1, RHS = 1 (1+1)//2
= 1 (2)//2
= 2//2
= 1
Induction:
For p (m) statement is true

So,
1+2+3+ ……… +m =m (m+1)//2 is true

For (m+1)
1+2+3+…. + (m+1) = (m+1) (m+1+1)//2
1+2+3+…. + (m+1) = (m+1) (m+1+1)//2

Subtract both sides (m+1)
1+2+3+…..+m= (m+1) (m+2)//2-(m+1)
= m2+2m+m+2-2m-2 //2
= m2+m //2
= m (m+1) //2

So the statement is true for p (m+1)
So, we can say Conclude: 1+2+3+…+n=n (n+1)//2 is true

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