U is the set of positive integers less than or equal to 30. A is the set of natural numbers that are multiples of 5 in U. B is the subset of all of the even integers in U.
a) Find n(A U B)
b) Find n(A intersect B)
c) Find n(A U B' )
d Find n(B intersect A')
Thank you so much!!!!!
To solve these problems, we need to first understand the given sets.
Given:
U: the set of positive integers less than or equal to 30.
A: the set of natural numbers that are multiples of 5 in U.
B: the subset of all even integers in U.
Now let's solve each part of the question:
a) Find n(A U B):
To find n(A U B), we need to find the total number of elements in the union of sets A and B.
1. Set A consists of natural numbers that are multiples of 5 in U.
So, the numbers included in A are {5, 10, 15, 20, 25, 30}. (Total: 6)
2. Set B consists of even integers in U.
The even numbers in U are {2, 4, 6, ..., 30}. (Total: 15)
To find the union (A U B), we need to combine the numbers from both sets, excluding any duplicates.
A U B = {2, 4, 5, 6, ..., 10, 15, ..., 20, ..., 25, 30}
Counting the numbers, we find that n(A U B) = 25.
b) Find n(A intersect B):
To find n(A intersect B), we need to find the number of elements that are present in both sets A and B.
Set A consists of multiples of 5 in U, and set B consists of even integers in U.
Multiples of 5 in U: {5, 10, 15, 20, 25, 30} (Total: 6)
Even integers in U: {2, 4, 6, ..., 30} (Total: 15)
The common elements in both sets are {10, 20, 30}.
Counting the numbers, we find that n(A intersect B) = 3.
c) Find n(A U B'):
This represents the union of set A and the complement of set B. The complement of B, denoted as B', consists of all elements in U that are not in B.
To find B', we need to find the odd integers in U.
Odd integers in U: {1, 3, 5, 7, ..., 29}
A U B' means combining the elements from set A and the elements from set B' while excluding any duplicates.
A: {5, 10, 15, 20, 25, 30}
B': {1, 3, 5, 7, ..., 29}
A U B' = {5, 10, 15, 20, 25, 30, 1, 3, 7, 9, ..., 29}
Counting the numbers, we find that n(A U B') depends on the range of odd numbers in U.
d) Find n(B intersect A'):
This represents the intersection of set B and the complement of A. The complement of A, denoted as A', consists of all elements in U that are not in A.
To find A', we need to remove multiples of 5 from the set U.
A: {5, 10, 15, 20, 25, 30}
A' is the set obtained by removing these multiples of 5 from U: {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, ..., 29}
(Note: We don't include 0 as it is not a positive integer.)
Now, we find the intersection of set B and set A'.
B: {2, 4, 6, ..., 30}
A': {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, ..., 29}
The common elements in both sets are {2, 4, 6, ..., 30}.
Counting the numbers, we find that n(B intersect A') depends on the range of even numbers in U.
In order to provide precise answers for parts c and d, we need to know the specific ranges of odd and even numbers in U. If these ranges are not mentioned, it would be difficult to find the exact counts.
well, work it out
U = {1,2,3,...,28,29,30}
A = {5,10,15,20,25,30}
B = {2,4,6,...,28,30}
Now can you find the unions and intersections? You must have some ideas here.