Consider the universal set U = (x is an element of N/3 < x < 13), and the subsets A= (multiples of 3) and B =(4,6,12)
a. List the elements of the following set.
A
b. List the elements of the following set.
*the intersection of A and B'
c. write down one element of (A u B)'
d. One of the elements of the statements in the table below is false. Indicate with an X which statement is false. Give a reason for your answer.
n (A u B) = 4 ____________
15 is an element of A' ____________
A is a subset of A u B ____________
A = {6,9,12}
Now the rest should be easy, if you understand anything at all about set operations.
a. To list the elements of set A, we need to find all the multiples of 3 between 3 and 13 (inclusive). The multiples of 3 in this range are 3, 6, 9, and 12. Therefore, the elements of set A are (3, 6, 9, 12).
b. The intersection of A and B' means we need to find the elements that are present in both set A and the complement of set B. The complement of set B (B') refers to all the elements in the universal set U that are not present in set B. In this case, the universal set U is all natural numbers between 3 and 13 (excluding 3 and 13).
The elements of set B are (4, 6, 12). The complement of set B (B') will include all the numbers in the range 3 to 13 that are not present in set B. So, B' would be (3, 5, 7, 8, 9, 10, 11, 13).
The intersection of A and B' refers to the elements that are common to both sets A and B'. From the elements listed above, we can see that the numbers 6 and 12 are present in both sets A and B'. Therefore, the elements of the intersection of A and B' are (6, 12).
c. (A u B)' represents the complement of the union of sets A and B. To find an element of this set, we need to consider all the elements in the universal set U that are not present in set A or set B.
The elements of set A are (3, 6, 9, 12) and the elements of set B are (4, 6, 12). The union of A and B (A u B) would include all the unique elements present in both sets, giving us (3, 4, 6, 9, 12).
The complement of (A u B) would include all the elements in the universal set U that are not present in (3, 4, 6, 9, 12). Considering the range of the universal set U (3 < x < 13), an element that is not present in (3, 4, 6, 9, 12) could be, for example, 5.
So, one element of (A u B)' is 5.
d. Let's check each statement in the table:
1. n(A u B) = 4: This statement represents the number of elements in the set A union B. From our previous calculations, we found that set A union B is (3, 4, 6, 9, 12), which contains 5 elements. Therefore, this statement is false.
2. 15 is an element of A': Since set A consists of multiples of 3 between 3 and 13 (excluding 3 and 13), the number 15 is not in set A. Therefore, this statement is true.
3. A is a subset of A u B: Set A consists of multiples of 3 between 3 and 13 (excluding 3 and 13). When we look at the elements of A union B, we can see that all elements of set A are included in set A union B. Therefore, set A is indeed a subset of A union B. This statement is true.
So, the false statement is: n(A u B) = 4. The reason for this is that the set A union B contains 5 elements, not 4.