Determine if the following proofs are valid or not. If they are not valid, find

Question

Determine if the following proofs are valid or not. If they are not valid, find a flaw in the proof o Statement: 13 o Proof: I claim that 1 3. Subtracting2 from both sides gives-1. Finally, squaring both sides gives the true statement that 11. This completes the proof o Statement: If n is even, then 5n is even. o Proof: Let n be any integers, and 2. suppose that n is even. Then we have that n 2k for some integers k. So, 5n - 5(2k) -2(5k). Since 5k is an integer, we conlude that 5n must be even. This completes the proof Statement: If ab is an even number, the 3. a or b is even o Proof: Assume that a or b is ever Without loss of generality, let's assume that it is a (a similar proof is given if b were even). So, a - 2k for some integer k. This gives that ab (2k)b -2(kb). Therefore ab is even. This completes the proof

Explanation / Answer

Answer is as follows :

1.

we have 1 = 3

Prove according to given info :

1 - 2 = 3 - 2 (subtract 2 frrom oth sides)

- 1 = 1

Square both sides, we get 1=1, that violates the given statement. so statement is invalid or we can say that proof is invalid..

2.

We have n is even.

we have n = 2k, where k is any integer

let k = 1

so n = 2(1) = 1

according to given info

5n = 5(2k) = 2(5k).

So let us put values

5n = 5(2(1) = 2(5(1)

5n = 10 = 10,

this conclude that 5n is even no.

So the proof is valid.

3.

we have ab is even.

So for prove

suppose a is even , so a = 2k (say k = 1), so a = 2(1) = 2

and let b = 1.

So accordingly

ab = 2k(b) = 2(kb)

ab = 2(1)(1) = 2(1 (1))

ab = 2 = 2

So we get ab = 2 i.e. even, so statement and proof are valid.

if there is any query please ask in comments...

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