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The Venn diagram represents the relationship among subsets of real numbers: rational numbers, irrational numbers, integers, whole numbers, and natural numbers. Use the Venn diagram to determine which statements are true. (Choose 3)

A Some irrational numbers are integers.
B All rational numbers are integers.
C Some rational numbers are whole numbers.
D Some, but not all, natural numbers are rational numbers.
E All rational and irrational numbers are real numbers.
F Rational numbers and irrational numbers have no numbers in common.

1 answer

The correct statements are:

C Some rational numbers are whole numbers.
D Some, but not all, natural numbers are rational numbers.
E All rational and irrational numbers are real numbers.

Explanation:

A Some irrational numbers are integers: This statement is false. In the Venn diagram, the sets for irrational numbers and integers do not overlap, indicating that there are no numbers that are both irrational and integers.

B All rational numbers are integers: This statement is false. The set for rational numbers includes both integers and non-integers, so not all rational numbers are integers.

C Some rational numbers are whole numbers: This statement is true. The set for whole numbers is a subset of the set for rational numbers, indicating that there are rational numbers that are also whole numbers.

D Some, but not all, natural numbers are rational numbers: This statement is true. The set for natural numbers is a subset of the set for rational numbers, indicating that there are natural numbers that are also rational numbers. However, not all natural numbers are rational numbers, as there are irrational numbers as well.

E All rational and irrational numbers are real numbers: This statement is true. In the Venn diagram, the sets for rational, irrational, and real numbers overlap, indicating that all rational and irrational numbers are also real numbers.

F Rational numbers and irrational numbers have no numbers in common: This statement is false. The sets for rational and irrational numbers overlap, indicating that there are numbers that are both rational and irrational. However, this overlap is only for some numbers, as there are also rational and irrational numbers that are not shared.