Write the following statement using logical symbols.
Let x be a positive integer. Then x = 2, or x is even and x > 2, or x is odd and prime, or
x is odd and composite.

The statement can be written in logical symbols as follows:

Let \( x \) be a positive integer.

\[
x = 2 \ \lor \ \left( x \text{ is even} \land x > 2 \right) \ \lor \ \left( x \text{ is odd} \land x \text{ is prime} \right) \ \lor \ \left( x \text{ is odd} \land x \text{ is composite} \right)
\]

Breaking down each condition:
- \( x = 2 \)
- \( \left( x \text{ is even} \land x > 2 \right) \) can be expressed as \( \left( x \mod 2 = 0 \land x > 2 \right) \)
- \( \left( x \text{ is odd} \land x \text{ is prime} \right) \) can be expressed as \( \left( x \mod 2 \neq 0 \land \text{Prime}(x) \right) \)
- \( \left( x \text{ is odd} \land x \text{ is composite} \right) \) can be expressed as \( \left( x \mod 2 \neq 0 \land \neg \text{Prime}(x) \land x > 1 \right) \)

Thus, the full logical expression is:

\[
x = 2 \ \lor \ \left( x \mod 2 = 0 \land x > 2 \right) \ \lor \ \left( x \mod 2 \neq 0 \land \text{Prime}(x) \right) \ \lor \ \left( x \mod 2 \neq 0 \land \neg \text{Prime}(x) \land x > 1 \right)
\]

Would you like any further details or clarification?

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Here are 5 related questions you might explore:
1. What is the logical symbol for "if and only if" and how does it apply to this statement?
2. How would you express the condition for \( x \) being a prime number using logical symbols?
3. Can a number be both even and odd? How would that be expressed in logic?
4. How does modular arithmetic help in determining whether a number is even or odd?
5. What are the logical conditions for a number being composite?

**Tip:** Using modular arithmetic (like \( x \mod 2 = 0 \)) is a simple way to check if a number is even or odd in logic.

Write the following statement using logical symbols. Let x be a positive integer. Then x = 2, or x is even and x > 2, or x is odd and prime, or x is odd and composite.

Learn how to express properties of integers using logical symbols. This example covers the logical conditions for an integer being even, odd, prime, or composite, using modular arithmetic and prime definitions. Suitable for students in Grades 10-12.

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