Prove forall x elementof z, x^2 - 5x - 1 is odd.

Answer with Step-by-step explanation:

Let Z be a set of integers

We have to prove that is odd for all value of x Z.

There are two cases

1.When x is an odd integer number

2.when x is even integer number

1.When x is an odd integer  number

When x is positive odd integer

Then  is positive odd number (because square of odd positive number is always positive an odd number )

5x is also odd number

-5x -1 = -even number

Suppose x=3

Then -15-1=-16

Odd number - even number=Odd number

Hence , is an odd number .

When x is negative odd number

Then  x square is positive odd number  and 5x is negative odd term term

Therefore,odd number + odd number -1=Even  number -1=Odd number

Hence is an odd number.

2.When  x is an even number

When x is positive even number

x square is positive even number  and 5x is positive even number

Even number -Even number -1 =Even number -odd number =Odd number

Suppose x=4

=16-20-1=-5=Odd number

When x is negative even number

Then x square is positive even number  and 5x is negative even number

=Even number +Even number -1=Even number -1=Odd number

Hence, for all elements of Zis odd.

its 8

step-by-step explanation: