Discrete Mathematic Proof Prove that if x^3 is irrational then x is also irratio

Question

Discrete Mathematic Proof

Prove that if x^3 is irrational then x is also irrational. First write a proof on your own and then put the following statements into order to obtain the proof. Put N next to the 4 statements that should not be used in the proof. Thus, x^3 = c/f, and by definition of rational x^3 is rational. By definition of rational, there exist integers a, b notequalto 0 such that x = a/b. By definition of rational, there exist real numbers a, b notequalto 0 such that x = a/b. Since a and b are integers, e = a^3 and f = b^3 is also an integer. Therefore, if x is rational then x^3 is rational, or equivalently if x^3 is irrational then x is irrational. Since the cube of a fraction is a fraction x^3 is rational. Suppose x is an arbitrary rational number. By definition of rational, there exist a b notequalto 0 such that x = a/b. Suppose x^3 is an arbitrary irrational number. Then, x^3 = a^3/b^3.

Explanation / Answer

We use a proof by contrapositive : If x is rational , then x3 is rational.

If x is rational , then we may write x = a/b, where a,b are integers and b not equal to 0. Then x3= a3/b3 and b3 not equal to 0( b not equal to 0).

Clearly a3 , b3 are also integers this proves x3 is rational.

The converse is : If x is irrational , then x3 is irrational. This i easily seen to be false ,

Counterexample : if x = 32 which is irrational but x3= 2 is rational.

This is the proof. You can put statements in order , after see this proof.

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