Question

which venn diagram correctly represents the relationship between rational numbers and irrational numbers?
, because some irrational numbers are also rational numbers
real numbers
rational numbers
irrational numbers
, because rational numbers and irrational numbers have no numbers in common
real numbers
rational numbers
irrational numbers
, because all rational numbers are also irrational numbers
real numbers
irrational numbers
rational numbers
, because all irrational numbers are also rational
real numbers
rational numbers

Explanation:

Step1: Recall number set relationships

Real numbers include rational and irrational numbers. Rational numbers (\( \mathbb{Q} \)) are numbers that can be expressed as \( \frac{p}{q} \) (\( p,q \in \mathbb{Z}, q
eq 0 \)), like \( 2, \frac{1}{2} \). Irrational numbers (\( \mathbb{I} \)) are non - repeating, non - terminating decimals, like \( \sqrt{2}, \pi \). By definition, a number cannot be both rational and irrational. So \( \mathbb{Q} \cap \mathbb{I}=\varnothing \), and \( \mathbb{Q}\cup\mathbb{I}=\mathbb{R} \) (real numbers).

Step2: Analyze each Venn diagram

• First diagram: Incorrect, as it implies irrational numbers are a subset of rational numbers, which is false.
• Second diagram: Incorrect, as it implies rational numbers are a subset of irrational numbers, which is false.
• Third diagram: Correct, as rational and irrational numbers are disjoint (no overlap) and together make up real numbers.
• Fourth diagram: Incorrect, as it implies there is an overlap between rational and irrational numbers, which is false. The reason for the third diagram is "because rational numbers and irrational numbers have no numbers in common" (since they are mutually exclusive subsets of real numbers).

Answer:

The Venn diagram with two separate ovals (rational numbers and irrational numbers) inside the rectangle of real numbers, with the reason "because rational numbers and irrational numbers have no numbers in common".