Solved on Dec 10, 2023

Prove the statement "if 2x=32^{x} = 3, then xx is not rational" is logically equivalent to the original statement "if xx is rational, then 2x32^{x} \neq 3".

STEP 1

Assumptions
1. We are dealing with a statement and its contrapositive.
2. The original statement is "if xx is rational then 2x32^{x} \neq 3".
3. The contrapositive of a statement "if P then Q" is "if not Q then not P".
4. We need to formulate the contrapositive of the given statement.

STEP 2

Identify the original statement's components.
Let P be the statement "xx is rational" and Q be the statement "2x32^{x} \neq 3".

STEP 3

Formulate the contrapositive of the original statement using the components identified in STEP_2.
The contrapositive is "if not Q then not P".

STEP 4

Translate "not Q" and "not P" into mathematical terms based on the original statement.
"Not Q" translates to "2x=32^{x} = 3" and "not P" translates to "xx is not rational".

STEP 5

Combine the translations from STEP_4 to form the contrapositive statement.
The contrapositive statement is "if 2x=32^{x} = 3 then xx is not rational".
The contrapositive of "if xx is rational then 2x32^{x} \neq 3" is indeed "if 2x=32^{x}=3 then xx is not rational".

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