Question

the venn diagram shows three types of numbers: odd (o), even (e), and prime (p). which is represented by ∅? options: o ∪ p, e ∩ p, o ∪ e, e ∩ o

Explanation:

1. Recall the definitions: An odd number (O) is not divisible by 2, and an even number (E) is divisible by 2. So a number cannot be both odd and even at the same time.
2. Analyze each option:

• \( O \cup P \): Contains odd numbers and prime numbers (e.g., 3 is in both O and P, 2 is in P and E, etc.), so not empty.
• \( E \cap P \): The intersection of even and prime numbers includes 2 (since 2 is even and prime), so not empty.
• \( O \cup E \): Contains all odd and even numbers (since every integer is either odd or even), so not empty.
• \( E \cap O \): The intersection of even and odd numbers. Since a number can't be both even (divisible by 2) and odd (not divisible by 2), there are no numbers in this set, so it's the empty set \( \varnothing \).

Answer:

\( E \cap O \)