Question

(1 point) Consider the universal set U={1,2,3,4,5,6,7,8,9,10}, define the set A be the even numbers, the set B be the odd numbers, and the set C={4,5,6}. Complete the following exercises in set notation.

(a) A U C
(b) B n C
(c) A n B
(d) B - C

Answer 1

The operations with sets result in A U C = {2, 4, 6, 8, 10}, B n C = {5}, A n B = {}, and B - C = {1, 3, 7, 9}. These results show the union, intersection, and set difference applied to sets A, B, and C with respect to the universal set U.

Step-by-step explanation:

To answer the student's question regarding sets, we need to understand the operations involved like union (U), intersection (n), and set difference (-). Given the universal set U={1,2,3,4,5,6,7,8,9,10}, where set A is the even numbers, set B is the odd numbers, and set C={4,5,6}:

• (a) A U C means the union of sets A and C, which includes all elements that are in A or C or both. Since A is the set of even numbers and C includes the even number 4 and 6, the result would simply be the set of all even numbers in U, as 6 adds no new element to A. Therefore, A U C = {2, 4, 6, 8, 10}.
• (b) B n C represents the intersection of sets B and C, meaning any numbers that are both odd and in set C. Since 5 is the only number that is odd in C, the result would be B n C = {5}.
• (c) A n B denotes the intersection of sets A and B, which would be any number that is both even and odd. However, no number can be both even and odd, so the result is an empty set, A n B = {}.
• (d) B - C is the set difference of B and C, which means it includes all elements of B that are not in C. Since B is the set of odd numbers and C includes the odd number 5, we remove 5 from B. The result is B - C = {1, 3, 7, 9}.

Answer 2

a) AUC = {2,4,6,8,10}

b) BnC = {}

c) AnB = {}

d) B-C = B = {1,3,5,7,9}

Step-by-step explanation:

The set A is the even numbers, those that are divisible by two.

So A = {2,4,6,8,10}

B is the odd numbe.rs. An odd number is a number that is not divisible by two.

So B = {1,3,5,7,9}.

C = {4,5,6}, as the problem states

a) The union of sets is a set containing all elements that are in at least one of the sets. So the union of A and C is a set that contains all elements that are in at least one of A or C.

So AUC = {2,4,6,8,10}.

b) The intersection of two sets consists of all elements that in both sets. So, the intersection of B and C is the set that contains all elements that are in both B and C.

There are no elements that are in both B and C, so the intersection is an empty set

BnC = {}

c) Same explanation as b), there are no elements that are in both A and B, so another empty set.

AnB = {}

d) The difference of sets B and C consists of all elements that are in B and not in C. We already have in b) that BnC = {}, so:

B-C = B = {1,3,5,7,9}