Given an implication "If P then Q" (P implies Q), the contrapositive is "If not Q then not P".
These statements are logically equivalent.
The only situation where "P implies Q" is false is when P is true and Q is false.
Proving the contrapositive can simplify proofs.
Example 1: Multiples of 3 and Sum of Consecutive Integers
Statement: Every multiple of 3 (in integers) is the sum of 3 consecutive integers.
Consecutive Integers: Form like x, x+1, x+2.
Predicate Logic Translation:
For all x in integers, if x is a multiple of 3, then x can be expressed as the sum of three consecutive integers.
Contrapositive Proof:
Translate the statement using logic quantifiers.
For all x not being sum of three consecutive numbers implies x is not a multiple of 3.
Simplify expressions to derive x is not equal to 3z + 3.
Conclude that x cannot be written in the form 3y if it doesn't fit 3z + 3.
Conclusion: Sometimes easier to prove the contrapositive.
Example 2: Rational and Irrational Numbers
Statement: If a real number x is irrational, then 1/x is irrational.
Predicate Logic Translation:
For all x in reals, if x is not rational, then 1/x is not rational.
Contrapositive Proof:
The contrapositive: If 1/x is rational, then x is rational.
Assume 1/x is a rational number, expressed as a ratio of integers a/b.
Derive that x can be expressed as b/a, hence x is rational.
Conclusion: Proving the contrapositive can be advantageous, especially when dealing with irrational numbers which are defined by the absence of integer expressions.
Key Takeaways
Proving the contrapositive can be a powerful tool in logic and proofs.
It is often easier to demonstrate a contrapositive than the original statement.
Practice helps in identifying when the contrapositive might simplify a proof.
Understanding rational versus irrational definitions aids in logical proofs.