Question

Given Prime(p) : p > 1 ∧ (∀d ∈ ℕ, d | p ⇒ (d = 1 ∨ d = p)), where p is a prime number. What does this expression signify about p?

a) p is not a prime number
b) p is a composite number
c) p is a perfect number
d) p is a prime number

Answer 1

The expression defines a prime number as one greater than 1 whose only divisors are 1 and itself. The symbol ∀ signifies 'for all' and d | p means 'd divides p'. The correct answer to the question is that p is a prime number.

Step-by-step explanation:

The expression given in the question defines the mathematical concept of a prime number. It states that p is a prime number if it is greater than 1 and the only divisors of p are 1 and p itself. Specifically, the symbol ∀ denotes 'for all' and d | p means 'd divides p'. Therefore, if every divisor d of p is such that d equals either 1 or p, then p meets the definition of a prime number.

In terms of the options provided, the correct answer is (d) p is a prime number, since that is precisely what the expression is defining. In contrast, a composite number would have divisors other than just 1 and itself, and a perfect number is a different concept entirely, where a number is equal to the sum of its proper divisors.