Consider the statement, “For all natural numbers n,n, if nn is prime, then nn is solitary.”...

Consider the statement, “For all natural numbers n,n, if nn is prime, then nn is solitary.” You do not need to know what solitary means for this problem, just that it is a property that some numbers have and others do not.

Write the converse and the contrapositive of the statement, saying which is which. Note: the original statement claims that an implication is true for all n,n, and it is that implication that we are taking the converse and contrapositive of.
Write the negation of the original statement. What would you need to show to prove that the statement is false?
Even though you don't know whether 10 is solitary (in fact, nobody knows this), is the statement “if 10 is prime, then 10 is solitary” true or false? Explain.
It turns out that 8 is solitary. Does this tell you anything about the truth or falsity of the original statement, its converse or its contrapositive? Explain.
Assuming that the original statement is true, what can you say about the relationship between the set PP of prime numbers and the set SS of solitary numbers. Explain.

Solutions

Expert Solution

The fact that 8 is solitary does not violate the original statement or its contrapositive. So we cannot conclude that the statements are FALSE. However, we also cannot say that this means the statements are true, because the natural number 8 does not violate the statements does not mean that no other natural number will.

Colby Messinger answered 1 year ago

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