Show that each conjecture is false by finding a counterexample:

1) If 1-y > 0, then 0 < y <1.

AND. . .

2)For any real number x, x cubed(to the power of 3) is greater than or equal to x squared(to the power of 2).

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1) If 1-y > 0, then 0 < y <1.

The conjecture is false.

Counterexample:

We take y= -15. Then 1-y =1- -15 = 16 >0. So if condition is satisfied. But the condtion 0 < y < 1 condtion is not satisfies as - 15 does not belong to (0 , 1) interval. So the cojecture is not true.

AND. . .

2)For any real number x, x cubed(to the power of 3) is greater than or equal to x squared(to the power of 2).

The conjectre if x is a real number, then x^3 > x^2.

The conjecture is false.

Couter example:

Let x = -5, then x^2 = (-5)^2 = 25 .

x^3 = (5) ^3 = -125.

So x^3 > x^2 is false , as (-5)^3 = < (-5)^2 .

Let x = 0.9. Then x^2 = 0.81 and x^3 = (0.9)^3 = 0.721. So in this example x^3 < x^2 as against the conjecture. So the conjecture is false.

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