Contrapositive of Multiples of 6

Problem Statement

Consider the statement: ``For all integers $n$, if $n$ is a multiple of 6, then $n$ is a multiple of 2.''

Which of the following is the contrapositive of this statement?

• There exists an integer $n$ such that $n$ is a multiple of 6 and not a multiple of 2.
• There exists an integer $n$ such that $n$ is a multiple of 2 and not a multiple of 6.
• For all integers $n$, if $n$ is not a multiple of 2, then $n$ is not a multiple of 6.
• For all integers $n$, if $n$ is not a multiple of 6, then $n$ is not a multiple of 2.

Hints

Attempt the proof independently first. Focus on the key theorem, algebraic transformation, or contradiction setup that links the hypothesis to the target conclusion.

Solutions

Logical Analysis of Contrapositive

Step 1: Identify the logical structure

The original statement has the form:

Where:

• $P(n)$: ``$n$ is a multiple of 6''
• $Q(n)$: ``$n$ is a multiple of 2''

Step 2: Apply contrapositive definition

The contrapositive of $P \implies Q$ is $\neg Q \implies \neg P$.

Key: The universal quantifier (``for all'') remains unchanged.

Step 3: Negate each part

• $\neg Q(n)$: ``$n$ is NOT a multiple of 2''
• $\neg P(n)$: ``$n$ is NOT a multiple of 6''

Step 4: Construct the contrapositive

In words: ``For all integers $n$, if $n$ is not a multiple of 2, then $n$ is not a multiple of 6.''

This matches Option C.

Analysis of incorrect options:

• Option A: Negation of original ($\exists n: P(n) \land \neg Q(n)$), not contrapositive
• Option B: Negation of converse
• Option D: Inverse ($\neg P \implies \neg Q$), not contrapositive

Answer: C}

Takeaways

• Contrapositive Form: $P \implies Q$ has contrapositive $\neg Q \implies \neg P$ (swap and negate both parts)
• Logical Equivalence: A statement and its contrapositive are logically equivalent (same truth value)
• Quantifiers Unchanged: Universal quantifier (``for all'') stays when forming contrapositive
• Common Errors: Inverse ($\neg P \implies \neg Q$) and converse ($Q \implies P$) are NOT equivalent to original

Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills:

• HSC Proofs: https://vumaths.com/booklets/hsc-proofs/
• HSC Integrals: https://vumaths.com/booklets/hsc-integrals/
• HSC Functions: https://vumaths.com/booklets/hsc-functions/

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