Math Problem Statement

Express the set of even natural numbers in set-builder notation.

Solution

The question asks to express the set of even natural numbers in set-builder notation.

Let’s go through the options:

  • A: E={xxN and x is odd}E = \{ x | x \in \mathbb{N} \text{ and } x \text{ is odd} \}
    This represents the odd natural numbers, which is not correct.

  • B: E={xx is an integer and x is odd}E = \{ x | x \text{ is an integer and } x \text{ is odd} \}
    This represents the odd integers, which is also incorrect.

  • C: E={xxN}E = \{ x | x \in \mathbb{N} \}
    This represents all natural numbers, not specifically the even ones, so this is incorrect.

  • D: E={xxN and x is a multiple of 8}E = \{ x | x \in \mathbb{N} \text{ and } x \text{ is a multiple of 8} \}
    This represents the natural numbers that are multiples of 8, not all even numbers. Hence, it’s incorrect.

  • E: E={xxN and x is even}E = \{ x | x \in \mathbb{N} \text{ and } x \text{ is even} \}
    This is correct because it specifies the natural numbers that are even.

  • F: E=E = \emptyset
    This represents an empty set, which is not correct.

Thus, the correct answer is E: E={xxN and x is even}E = \{ x | x \in \mathbb{N} \text{ and } x \text{ is even} \}.

Would you like further details or explanations on any part of the question? Here are some related questions for deeper understanding:

  1. What are the natural numbers (N\mathbb{N}) and how are they defined?
  2. How do we define even numbers in set theory?
  3. Can you describe how set-builder notation works with other types of numbers (e.g., odd numbers)?
  4. What is the difference between natural numbers and integers in set theory?
  5. How do we express other specific subsets of numbers, like prime numbers, using set-builder notation?

Tip: Remember that set-builder notation is a shorthand way of defining sets by specifying the properties that the members of the set must satisfy!

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Math Problem Analysis

Mathematical Concepts

Set Theory
Natural Numbers
Even Numbers

Formulas

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Theorems

Definition of natural numbers
Definition of even numbers in set theory

Suitable Grade Level

Grades 6-8