Question

Translate each of the following sentences into symbolic logic. If \(x\) is prime, then \(\sqrt{x}\) is not a rational number.

Short answer

The symbolic logic of the statement 'If \(x\) is prime, then \(\sqrt{x}\) is not a rational number' is \(p→q\) where p denotes 'x is prime' and q denotes '\(\sqrt{x}\) is not a rational number'

Step-by-step solution

Identify two propositions

Identify the two propositions in the sentence. Here, 'x is prime' and '\(\sqrt{x}\) is not a rational number' are two propositions that formulate the complete logic statement.

Define propositional variables

Assign propositional variables to two identified propositions. Let's define p: 'x is prime' and q: '\(\sqrt{x}\) is not a rational number'

Translate into symbolic logic

The sentence 'If p, then q' can be translated into logic as 'p implies q' or 'p→q'. In this case, 'If x is prime, then \(\sqrt{x}\) is not a rational number' becomes 'p→q'

Key concepts

Propositional Variables

In the world of symbolic logic, propositional variables are the basic building blocks. They represent statements that can be either true or false but cannot express more complex ideas such as 'sometimes' or 'maybe'. For instance, in the statement 'If x is prime, then \(\sqrt{x}\) is not a rational number', we have two propositions: \( p \) representing 'x is prime' and \( q \) representing '\(\sqrt{x}\) is not a rational number'.

Propositional variables make logical expressions precise by allowing us to focus on the truth value of entire statements without getting mired in the natural language complexities. They are the placeholders in an abstract mathematical world where we are concerned only with the logical structure rather than content of propositions.

Logical Implication

The concept of logical implication is a cornerstone of propositional logic. It's a way to express the idea that one statement leads to another. In logical terms, we often express this relationship using an arrow symbol, like \( p \rightarrow q \), which reads as 'if p, then q'. This is saying that if the first proposition (p) is true, then the second proposition (q) must also be true.

However, implication can be a tricky concept because it doesn't always follow the same intuition we have in everyday language. In logic, if the first proposition (p) is false, then the entire implication is considered true regardless of the truth value of the second proposition (q). This is because implication is only concerned with cases where the first statement is true and leads to the truth of the second one.

Prime Numbers

Within mathematics, prime numbers are those that are greater than 1 and have no divisors other than 1 and themselves. This means they cannot be formed by multiplying two smaller natural numbers together. Primes are the building blocks of number theory, as every number greater than 1 is either a prime or can be factored into primes.

Understanding prime numbers is crucial for various areas of mathematics including cryptography, where the difficulty of factoring large primes into their constituent prime factors underpins many encryption systems. In the context of our exercise, recognizing that a number is prime provides insight into its other properties, such as the nature of its square root.

Rational Numbers

The term rational numbers refers to any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, with the denominator \( q \) not equal to zero. All integers are rational numbers since any integer \( z \) can be written as \( \frac{z}{1} \).

Rational numbers include fractions like \( \frac{1}{2} \) or \( \frac{-3}{4} \), but not square roots of non-square numbers, like \( \sqrt{2} \), which are irrational. In our exercise, we are conceptualizing \( \sqrt{x} \) where \( x \) is prime, and part of the symbolic logic exercise was to understand that the square root of a prime number will not be a rational number.