Answer to Question #106199 in Discrete Mathematics for Justin Lee

Question

Give two examples of an implication which is true, but whose converse is not true. One example should be a real-world example, while the other should be an example from math involving the integers.

Answer 1

An example of a statement pertaining to the real-world with a false converse is as follows "If there is a tsunami, then there was an earthquake". This is true because a tsunami is caused by tectonic shifts. However, the converse is "If there was an earthquake, then there is a tsunami" which is untrue, for example, in California, which has many earthquakes, but no tsunamis.

An example in mathematics is as follows "Consider an integer n greater than 2. If n is prime, then n is odd". This statement is true, because all primes greater than 2 are odd (otherwise they would be divisible by 2). However the converse is not true, i.e. "If n is odd, then n is prime". A counterexample is 15, which is composite, even though it is odd.