Question

Translate each of the following sentences into symbolic logic. For every prime number \(p\) there is another prime number \(q\) with \(q>p\).

Short answer

\( \forall p \exists q (Prime(p) \land Prime(q) \land q>p) \)

Step-by-step solution

Definition of Variables

Let's denote the prime number greater than \(p\) by \(q\). So we have two variables \(p\) and \(q\) both of which are prime numbers.

Expression of relationship using logic

The statement says 'For every prime number \(p\) there is another prime number \(q\) with \(q>p\)' can be expressed symbolically as \( \forall p \exists q (Prime(p) \land Prime(q) \land q>p) \) where \( \forall \) represents 'for all', \( \exists \) represents 'there exists', and \( \land \) is the logical 'and' operator.

Key concepts

Prime Numbers

Prime numbers are the building blocks of whole numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, the numbers 2, 3, 5, and 7 are prime because they cannot be divided evenly by any other numbers besides 1 and their own value.

Understanding prime numbers is crucial in various areas of mathematics and science, especially in number theory and cryptography. In the context of our exercise, we consider the infinite nature of primes, meaning that for every prime number, there is always another prime number greater than it. This concept was proven by the famous mathematician Euclid and is known as Euclid's theorem.

This property plays an integral role in our symbolic logic exercise, where we showcase how logic can capture this infinite aspect of prime numbers.

Logical Operators

Logical operators are symbols or words used to connect propositions in logic to form more complex logical expressions. The basic operators include 'and' (\(\land\)), 'or' (\(\lor\)), 'not' (\(eg\)), 'if...then...' (\(\rightarrow\)), and 'if and only if' (\(\leftrightarrow\)).

In our exercise, the logical 'and' operator (\(\land\)) is used to connect multiple conditions that must all be true. It indicates that for a statement to be true, every individual proposition on either side of the 'and' operator must also be true. For example, in the expression \(Prime(p) \land Prime(q) \land q>p\), each component - \(Prime(p)\), \(Prime(q)\), and \(q>p\) - must be true for the entire expression to hold true.

The use of logical operators is essential in constructing precise and unambiguous statements in mathematics, computer science, and philosophy.

Quantifiers

Quantifiers are used in logic to indicate the extent to which a statement applies to the subjects of discourse. The two most common quantifiers are the universal quantifier (\(\forall\)), which symbolizes the phrase 'for all' or 'for every', and the existential quantifier (\(\exists\)), which indicates 'there exists' at least one.

In the statement from the exercise, 'For every prime number \(p\) there is another prime number \(q\) with \(q>p\)', we use both types of quantifiers. The universal quantifier \(\forall p\) tells us that the assertion we’re making applies to all prime numbers \(p\), without exception. Following this, the existential quantifier \(\exists q\) introduces the existence of at least one other prime number \(q\) that is greater than \(p\).

The correct use of quantifiers is vital for expressing the scope of propositions accurately. It allows mathematicians and logicians to define properties and set relationships across potentially infinite collections like prime numbers.