Question

1. The number x is multiplied by itself and the result y is less than the number x.

For example, x*x = x² = y and y < x. 

i.   State a set name in which numbers behave in such an assertion above.

ii.     State any number in the stated set in (a) above and verify the assertion. 

Answer 1

1. $y=x^2<x$

i.

$$x^2<x=>x^2-x<0=>x(x-1)<0$$

$$=>x\in(0, 1)$$

The opened interval $(0, 1).$

ii. Let $x=\dfrac{1}{2}.$ Then

$$y(\dfrac{1}{2})=(\dfrac{1}{2})^2=\dfrac{1}{4}<\dfrac{1}{2}=x$$