Question

(a) use the names of the sets to label the regions of the venn diagram.
names of the sets
integers
irrational numbers
natural numbers
whole numbers
(b) true or false?
statement\ttrue\tfalse
some irrational numbers are whole numbers.
all whole numbers are natural numbers.

Explanation:

Step1: Recall Number Set Hierarchy

Natural numbers ($\mathbb{N}$) are a subset of whole numbers ($\mathbb{W}$), whole numbers are a subset of integers ($\mathbb{Z}$), integers are a subset of rational numbers ($\mathbb{Q}$), and there's a region for non - rational (irrational) numbers (but here we focus on the given sets). The innermost circle (smallest) should be natural numbers, the middle circle (containing natural numbers) is whole numbers, the outer circle (containing whole numbers) is integers, and the right - hand region (outside integers) is rational numbers? Wait, no, rational numbers include integers, fractions, etc. Wait, correct hierarchy: Natural numbers ($\mathbb{N}$: 1, 2, 3,...) are a subset of Whole numbers ($\mathbb{W}$: 0, 1, 2, 3,...), Whole numbers are a subset of Integers ($\mathbb{Z}$:..., - 2, - 1, 0, 1, 2,...), and Integers are a subset of Rational numbers ($\mathbb{Q}$: numbers that can be expressed as $\frac{p}{q}$, $q
eq0$). So the Venn diagram has two nested circles (inner and middle) and an outer region? Wait, the diagram has a small circle inside a larger circle, and then a separate rectangle - like region. So:

• Innermost circle (smallest): Natural numbers (since natural numbers are a subset of whole numbers)
• Middle circle (containing natural numbers): Whole numbers (whole numbers are a subset of integers)
• Outer circle (containing whole numbers): Integers (integers are a subset of rational numbers? Wait, no, the remaining set is rational numbers? Wait, the sets given are integers, rational numbers, natural numbers, whole numbers. So the correct labeling:
• Innermost (small circle): Natural numbers
• Middle circle (enclosing natural numbers): Whole numbers
• Outer circle (enclosing whole numbers): Integers
• The region to the right (outside the integer circle): Rational numbers (but wait, rational numbers include integers. Wait, maybe the right region is for rational numbers that are not integers, and the integer circle is part of rational numbers.

Step2: Label the Regions

1. Innermost circle (smallest): Natural numbers
2. Middle circle (between innermost and outer): Whole numbers (since whole numbers include natural numbers and 0)
3. Outer circle (enclosing middle circle): Integers (since integers include whole numbers and negative numbers)
4. Region to the right (outside the integer circle): Rational numbers (rational numbers that are not integers, like $\frac{1}{2}$, $-\frac{3}{4}$, etc.)

For part (b):

• "Some rational numbers are whole numbers": Rational numbers include integers, and whole numbers are integers (including 0). So numbers like 2 (a whole number) is a rational number ($\frac{2}{1}$). So this statement is True.
• "All whole numbers are rational numbers": Whole numbers are integers, and integers can be written as $\frac{n}{1}$ (e.g., 0 = $\frac{0}{1}$, 5 = $\frac{5}{1}$). So all whole numbers are rational numbers. This statement is True.

Answer:

(a) Labeling: Innermost circle: Natural numbers; Middle circle: Whole numbers; Outer circle: Integers; Right region: Rational numbers.
(b) "Some rational numbers are whole numbers": True; "All whole numbers are rational numbers": True.