Question

1. State the converse, contrapositive, and inverse of the conditional statement “A positive integer is a prime only if it has no divisors other than 1 and itself”.

Answer 1

Explanation:

A contrapositive statement changes "if not p then not q" to "if not q to then, not p." The converse of the conditional statement is “If Q then P.” The contrapositive of the conditional statement is “If not Q then not P.” The inverse of the conditional statement is “If not P then not Q.” if p → q, p → q, then, ∼ q →∼ p ∼ q →∼ p

A Note about Notation: Be careful with the notation a|b

. This does not represent the rational number ab

. The notation a|b

represents a relationship between the integers a

and b

and is simply a shorthand for “ a

divides b

.” "Divides" as in a|b

is a relation (true or false), while "divided by" as in ab

or a/b

is an operation (results in a number).

The definition for “divides” can be written in symbolic form using appropriate quantifiers as follows: A nonzero integer m

divides an integer n

provided that (∃q∈Z)(n=m⋅q).