Question

Select the statements that are true based on the following given information.[ D = {x \mid x \text{ is a whole number}} ]

[ E = {x \mid x \text{ is a perfect square } < 36} ]
[ F = {x \mid x \text{ is an even number between 20 and 30}} ]The expression (D \cap E) is {1, 4, 9, 16}.
The expression (D \cap F) is {12, 14, 16, 18}.
The expression (D \cup (E \cap F)) is {all whole numbers}.
(E \cap F) is the empty set.
The expression (D \cap (E \cup F)) is 25.

Answer 1

Some provided statements about the intersection and union of sets D, E, and F are true, specifically that the intersection of D and E is {1, 4, 9, 16} and that E and F are likely disjoint sets.

Step-by-step explanation:

We need to verify which statements are true or false based on the given information about sets D, E, and F.

• The expression (D ∩ E) is {1, 4, 9, 16}. This is true because these are all of the perfect squares less than 36 that are also whole numbers.
• The expression (D ∩ F) is {12, 14, 16, 18}. This is false because the set F should only include even numbers between 20 and 30.
• The expression (D ∪ (E ∩ F)) is all whole numbers. This is false because the union of D with the intersection of E and F does not produce all whole numbers, only those that are in D, or are both perfect squares less than 36 and even numbers between 20 and 30.
• (E ∩ F) is the empty set. This is likely true because there are no perfect square even numbers between 20 and 30.
• The expression (D ∩ (E ∪ F)) is 25. This is false because 25 is not an even number between 20 and 30, and thus not in F; although it is a perfect square less than 36 and therefore in E, the intersection with D would not include 25 as it requires numbers to also be in F.