Prove an implication through its contrapositive while keeping the converse separate.
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Try
A direct route from an odd expression to an odd input can be awkward. Try asking what happens when the input is even. Sometimes the negated conclusion gives the simplest starting equation.
Predict
Predict the difference between a contrapositive and a converse. For “P implies Q,” the contrapositive is “not Q implies not P”; the converse is “Q implies P.” The first has the same truth value as the original implication; the second need not.
Experiment
Make a four-row truth table for P and Q in the logic investigation. Mark the one failing case of P implies Q. Then do the same for not Q implies not P. Their failure cases coincide. Repeat with the converse and observe the different failing row.
Understand
Worked argument: if 3n+1 is even for integer n, then n is odd. Prove the contrapositive: if n is even, write n=2k with integer k. Then 3n+1=6k+1=2(3k)+1 is odd. Thus an even n cannot produce an even 3n+1, establishing the desired implication.
An invalid proof instead assumes n odd and shows 3n+1 even. That proves the converse. In this particular example the converse also happens to be true, but proving it alone still leaves the requested implication unproved.
Apply
Label the original premise and conclusion before negating them. Over integers, not odd means even; over arbitrary real numbers this parity language is not appropriate. In an inequality, negate the boundary correctly: not greater than 2 means at most 2.
Continue
Depth: show by a truth table that an implication is also equivalent to “not P or Q.” Use that equivalence to explain why a false premise does not refute an implication. A converse may become a theorem with a separate proof; never obtain it by reversing an arrow without justification.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
What is the contrapositive of 'If 5n+2 is odd, then n is odd'?
For n=2k, write 5n+2=2(ak+1). What is a?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove the original implication by contrapositive for integer n.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For a real x, a learner proves 'x>2 implies x²>4' and concludes 'x²>4 implies x>2'. Diagnose with a counterexample.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
