To determine the contrapositive of a statement, we first need to understand what the contrapositive is. For a given implication statement "If [tex]\( p \)[/tex], then [tex]\( q \)[/tex]", the contrapositive is "If not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]".
The given statement [tex]\( p \)[/tex] is:
- If [tex]\( x \)[/tex] is divisible by 4, then [tex]\( x \)[/tex] is divisible by 2.
Let's analyze this statement and find its contrapositive step by step.
1. Identify [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in the original statement:
- [tex]\( p \)[/tex]: [tex]\( x \)[/tex] is divisible by 4.
- [tex]\( q \)[/tex]: [tex]\( x \)[/tex] is divisible by 2.
2. Write the contrapositive using the format: "If not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]":
- Not [tex]\( q \)[/tex]: [tex]\( x \)[/tex] is not divisible by 2.
- Not [tex]\( p \)[/tex]: [tex]\( x \)[/tex] is not divisible by 4.
3. Combine the negations into the contrapositive statement:
- If [tex]\( x \)[/tex] is not divisible by 2, then [tex]\( x \)[/tex] is not divisible by 4.
Now, let's match this with the given choices:
A. If [tex]\( x \)[/tex] is divisible by 2, then [tex]\( x \)[/tex] is divisible by 4.
- This is incorrect as it is not the contrapositive.
B. If [tex]\( x \)[/tex] isn't divisible by 4, then [tex]\( x \)[/tex] isn't divisible by 2.
- This is also incorrect as it suggests a different implication.
C. If [tex]\( x \)[/tex] isn't divisible by 2, then [tex]\( x \)[/tex] isn't divisible by 4.
- This is the correct contrapositive statement we derived.
D. If [tex]\( x \)[/tex] is divisible by 2, then [tex]\( x \)[/tex] isn't divisible by 4.
- This is incorrect as it negates the original statement.
E. If [tex]\( x \)[/tex] isn't divisible by 4, then [tex]\( x \)[/tex] is divisible by 2.
- This is incorrect as it incorrectly implies the conditions.
Therefore, the correct answer is:
C. If [tex]\( x \)[/tex] isn't divisible by 2, then [tex]\( x \)[/tex] isn't divisible by 4.
