Give a counterexample to each of the following to show that

Give a counterexample to each of the following to show that it is a false statement. If you think that the statement is actually true, give a brief explanation of why you think that no counterexample exists.

a. If sin (x) = 0, then cos (x)=1.

b. If x³ =x, then x²=1.

c. If x²=y², then x=y.

Solution

a) The given statement is False

Since sinx = 0, when the value of x is a multiple of n*pi, where n being a integer

sin(pi) = 0 and sin(2pi) = 0, but cos(pi) = -1 and cos(2pi) = 1

Hence x=pi is an contradiction to the above statement, since it satisfies that sinx = 0 but not cosx = 1

b) The given statement is also False

x^3 = x, when we are writing the statement then we are cancelling the x terms from both LHS and RHS, assuming that x is not zero

if x is not zero => x^2 = 1

Hence the given statement will be true if the number is any number except zero

c) x^2 = y^2 is also false

Since it can provide two solutions i.e. x=y and x=-y

Taking a counter example, assume x=1 and y=-1

Then we get x^2 = 1 and y^2 = (-1)^2 = 1

But the value of x is not equal to y, since x is equal to 1 and y is equal to -1