Problem 1 Find the error on the following proof that 2 = 1. Consider the equatio

Question

Problem 1

Find the error on the following proof that 2 = 1. Consider the equation a = b. Multiply both sides by a to obtain a2 = ab. Subtract b2 from both sides to get a2 b2 = ab b2. Now factor each side to get (a + b)(a b) = b(a b). Now divide each side by (ab) to get a+b = b. Finally, let a and b equal 1, which shows that 2 = 1.

Problem 2

Prove or disprove the following equality: ({a}{b}) = {a, b}

Problem 3

Prove that a language, L, is recursive iff there exists an algorithm that enu- merates L.

Explanation / Answer

Problem1.

Here in step(1) we have considered that a=b.

where as in step (4) we have (a+b)(a-b)=b(a-b)

and in step(5) we are dividing on bith sides with (a-b) which does not follow step(1)

why because in step(1) we have a=b ,the error in the above proof which finally shows that 2=1 arises in this step.

the reason is here we are dividing with (a-b) which is equal to '0'(zero) (since a=b according to step(1)) and

division by zero is undefined.consider the following explanation to clearly understand the concept of division by

zero.

EXPLANATION:

when you have 2 numbers X and Y ,you can only divide X by Y and get answer as another number ' X/Y' only when ' Y' is not equal to zero(0).

As you cannot get a solution when you divide a number by zero ,that's why " mathematicians say that division by zero is undefined."

Consider the following examples

Example(1): When you divide 6 by 2 what do you get?

(6/2)=3

                   You get answer as ‘ 3’,

  Why Because 2 times 3 is 6.

2*3 = 6

Example (2):What if you divide 20 by 5?

(20/5)=4

You get answer as ‘ 4’.

That's because 5*4 = 20.

   NOTE: x/y = z if and only if y*z = x

    Every time we divided, we got a unique answer, that is, we only got one number.

6/2 does not equal 4 because 2*4 = 8 and 8 is not equal to 6

    The only number that satisfies the equation 2*x = 6 is x = 3 ( x is unique).

Everything is clear till now. Now let’s see what happens when you start dividing by zero.

    What is 1 / 0 ?

Is it 1?

No. Because 0 * 1 = 1 is false.

Is it 2?

No. Because 0 * 2 = 1 is false.

    In fact, if you say 1/0 is the number x,

then x has tosatisfy the equation    0*x = 1,   which tells you that 0 = 1, which is an absolutely falsestatement. This means that there is no such number x, but x was 1/0, and so there is no number 1/0, and we say it's undefined.

   Similarly, we can show that

   2/0 is undefined,

   3/0 is undefined,

4/0 is undefined, and   so on...

Now, what about 0/0 ? What is it?

Is it equal to 0?

yes, because 0*0 = 0

    There's only one problem. we can show that 0/0 is also 1 !

Because 0*1 = 0.

we can also show that 0/0 is also 2 !   

    Because 0*2 = 0.

we can show that 0/0 is any number!

Because 0*(any number) = 0

Since we can show that 0/0 can be any number, we say that

0/0 is not unique, and for that reason it is also undefined.

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