Consider the statement, “For all natural numbers n, if n is prime, then n 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. (a) 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, and it is that implication that we are taking the converse and contrapositive of. (b) Write the negation of the original statement. What would you need to show to prove that the statement is false? (c) 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.
Consider the statement, “For all natural numbers n, if n is prime, then n 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.
(a) 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, and it is that implication that we are
taking the converse and contrapositive of.
(b) Write the negation of the original statement. What would you need to show to prove that the
statement is false?
(c) 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.
(d) 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.
(e) Assuming that the original statement is true, what can you say about the relationship between
the set P of prime numbers and the set S of solitary numbers. Explain.
Please just answer from letters D-E
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