Question

In: Advanced Math

Let S = {1,2,3,...,10}. a. Find the number of subsets of S that contain the number...

Let S = {1,2,3,...,10}.

a. Find the number of subsets of S that contain the number 5.

b. Find the number of subsets of S that contain neither 5 nor 6.

c. Find the number of subsets of S that contain both 5 and 6.

d. Find the number of subsets of S that contain no odd numbers.

e. Find the number of subsets of S that contain exactly three elements.

f. Find the number of subsets of S that contain exactly three elements, one of which is 3.

g. Find the number of subsets of S that contain exactly five elements, all of them even.

h. Find the number of subsets of S with exactly five elements, including 3 or 4 but not both.

Solutions

Expert Solution

a) Number of subsets that contain 5 - For each number 1, 2, ... 10 (except 5) we have 2 possibilities - it contains that number or not. Total such subsets is

b) Don't neither 5 nor 6 is as for each of the rest 8 numbers we have 2 possibilities - it contains that number or not.

c) Containing both - for the rest 8 numbers we have 2 possibilities - it contains that number or not. Again the number of such subsets is

d) There are 5 odd numbers - 1, 3, 5, 7, 9 so that we can 5 even numbers. Total number of subsets is

e) Exactly 3 elements is

f) 3 is always there. We have to choose 2 numbers from remaining 9 numbers so that is the number

g) There are exactly 5 even numbers in S. Thus there is only one subset of 5 elements with all of them even namely

h) Number of subsets with exactly 5 elements is

Number of subsets with 5 elements containing 3 is

Number of subsets with 5 elements containing 4 is

Number of subsets with 5 elements containing 3 and 4 is

Required number of subsets is

Hope this was helpful. Please do leave a positive rating if you liked this answer. Thanks and have a good day!

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

A)Let S = {1,2,3,...,18,19,20} be the universal set. Let sets A and B be subsets of...

A)Let S = {1,2,3,...,18,19,20} be the universal set. Let sets A and B be subsets of S, where: Set A={3,4,9,10,11,13,18}A={3,4,9,10,11,13,18} Set B={1,2,4,6,7,10,11,12,15,16,18}B={1,2,4,6,7,10,11,12,15,16,18} LIST the elements in Set A and Set B: { } LIST the elements in Set A or Set B: { } B)A ball is drawn randomly from a jar that contains 4 red balls, 5 white balls, and 9 yellow balls. Find the probability of the given event. Write your answers as reduced fractions or whole numbers. (a) PP(A...

Discrete Mathematics Find the number of subsets of S = {1, 2, 3, ... , 10}...

Discrete Mathematics Find the number of subsets of S = {1, 2, 3, ... , 10} that contain both 5 and 6. I know the answer is 2^8, however I have no idea how to go on to solve this problem.

Let S(n) be the number of subsets of {1,2,...,n} having the following property: there are no...

Let S(n) be the number of subsets of {1,2,...,n} having the following property: there are no three elements in the subset that are consecutive integers. Find a recurrence for S(n) and explain in words why S(n) satisfies this recurrence

Let a_n denote the number of sequences of 0's and 1's that do not contain two...

Let a_n denote the number of sequences of 0's and 1's that do not contain two consecutive 0's. Determine a_n.

Let S be a set of n numbers. Let X be the set of all subsets...

Let S be a set of n numbers. Let X be the set of all subsets of S of size k, and let Y be the set of all ordered k-tuples (s1, s2, , sk) such that s1 < s2 < < sk. That is, X = {{s1, s2, , sk} | si S and all si's are distinct}, and Y = {(s1, s2, , sk) | si S and s1 < s2 < < sk}. (a) Define a one-to-one correspondence f : X → Y. Explain...

Find a system of recurrence relations for the number of n-digit quaternary sequences that contain an even number of 2’s and an odd number of 3’s.

Find a system of recurrence relations for the number of n-digit quaternary sequences that contain an even number of 2’s and an odd number of 3’s. Define the initial conditions for the system. (A quaternary digit is either a 0, 1, 2 or 3)

Calculus #3: 1. a) Let A = (2,4,6),B = (1,2,3) and C = (5,5,5). Find point...

Calculus #3: 1. a) Let A = (2,4,6),B = (1,2,3) and C = (5,5,5). Find point D so that ABCD is a parallelogram. b). Two points X and Y are colinear if they lie on the same line. Are the points A = (3,6,−1), B = (2,0,3) and C = (−1, 3, −4) colinear? Justify your answer.

For the following exercises, find the number of subsets in each given set. {a, b, c, … , z}

For the following exercises, find the number of subsets in each given set.{a, b, c, … , z}

Let Ω be any set and let F be the collection of all subsets of Ω...

Let Ω be any set and let F be the collection of all subsets of Ω that are either countable or have a countable complement. (Recall that a set is countable if it is either finite or can be placed in one-to-one correspondence with the natural numbers N = {1, 2, . . .}.) (a) Show that F is a σ-algebra. (b) Show that the set function given by μ(E)= 0 if E is countable ; μ(E) = ∞ otherwise...

find a recurrence relation for the number of bit strings of length n that contain the...

find a recurrence relation for the number of bit strings of length n that contain the string 10. What are the initial conditions? How many bit strings of length eight contain the string 10

Subjects