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Solved on Dec 10, 2023

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

STEP 1

Assumptions
1. We are dealing with a statement and its contrapositive.
2. The original statement is "if $x$ is rational then $2^{x} \neq 3$ ".
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 " $x$ is rational" and Q be the statement " $2^{x} \neq 3$ ".

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 " $2^{x} = 3$ " and "not P" translates to " $x$ is not rational".

STEP 5

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

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