Question

Define the following sets. a = {x ∈ z: x is a multiple of 3} b = {3, 5, 7, 9} c = {2, 3, 4, 5} indicate whether each statement is true or false.

(a) |b| = |c|
(b) |a ∩ b| = |a ∩ c|

Answer 1

Set a represents all integers that are multiples of 3. The statement |b| = |c| is true because both have four elements. Assuming all multiples of 3 for set a, the statement |a ∩ b| = |a ∩ c| is false, as set b has two and set c has one element that is a multiple of 3.

Step-by-step explanation:

The definition of the sets given in the question is as follows:

• a = {x ∈ ℤ: x is a multiple of 3} represents all integers that are multiples of 3.
• b = {3, 5, 7, 9} is a set containing four specific integers.
• c = {2, 3, 4, 5} is another set containing four specific integers.

To evaluate the given statements:

1. |b| = |c| - This statement is true because both sets b and c contain four elements, so their cardinalities are equal.
2. |a ∩ b| = |a ∩ c| - To determine the truth of this statement we identify the common elements between set a and sets b and c respectively. Since set a includes all multiples of 3, it will have common elements with both sets b and c. However, without specifying which multiples of 3 are being considered, we cannot definitively compute the cardinalities of a ∩ b and a ∩ c. Assuming we take the definition of set a to include all possible multiples of 3, |a ∩ b| would be 2 (because 3 and 9 are multiples of 3 and are in set b) and |a ∩ c| would be 1 (since only the number 3 is a multiple of 3 in set c). Therefore, if we assume that set a contains all multiples of 3 and the sets b and c are as defined, the statement is false because |a ∩ b| ≠ |a ∩ c|.