3 questions if you can help me.
1. Calculate the number of subsets and the number of proper subsets for the set.
{x|x is a day of the week}
2. Let U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
C = {v, w, x, y, z}. List the elements in the set.
(A n B)u(A n C)
1. the answer is the "power set", i.e. the set of all possible subsets of the given set. The cardinality of the power set of A is equal to 2^(|A|).
For example:
A={1,2}
power set of A ={∅,1,2,{1,2}}
with 4 elements = 2^2.
so if {x|x is a day of the week},
then
n=|x|=7, and the power set of x has 2^7 members.
The number of proper subset is 1 less than 2^n.
2.
A∩B={q,s,y}
A∩C={w,y}
Can you find (A∩B)∪(A∩C)?
Sure, I can help you with your questions. Let's start with the first one:
1. To calculate the number of subsets and proper subsets for the set {x | x is a day of the week}, we need to understand the concept of subsets and proper subsets.
A subset is a set that contains some or all of the elements of the original set. A proper subset is a subset that contains some, but not all, of the elements of the original set.
In this case, the set {x | x is a day of the week} contains 7 elements (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday).
To calculate the number of subsets, we use the formula 2^n, where n is the number of elements in the original set. In this case, there are 7 elements, so there will be 2^7 = 128 subsets.
To calculate the number of proper subsets, we subtract 1 from the total number of subsets. So, there will be 128 - 1 = 127 proper subsets.
Now let's move on to the second question:
2. The given sets are:
U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
C = {v, w, x, y, z}
We need to find the elements in the set (A n B) u (A n C).
To find A n B, we need to find the elements that are common to sets A and B. In this case, the common elements are q, s, and y.
To find A n C, we need to find the elements that are common to sets A and C. In this case, the common elements are w and y.
So, (A n B) is {q, s, y} and (A n C) is {w, y}.
To find (A n B) u (A n C), we combine the elements from both sets without duplicating any elements. So, (A n B) u (A n C) is {q, s, y, w}.
Therefore, the elements in the set (A n B) u (A n C) are q, s, y, and w.
I hope this helps! Let me know if you have any further questions.