1. Suppose that C is generated from a set B= {a,b} by the binary operation f and the unitoperation g . List all the elements of C, . How many elements can they have C, y C,?Remember that C, is the set of all points x such that some construction sequence oflength n ends with x, and soB = CcC,cC; s..2. Prove that (A, →^4,) it is not a formula

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College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.5: Counting Principles

Problem 38SE: Suppose a set A has 2,048 subsets. How many distinct objects are contained in A?

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Suppose a set A has 2,048 subsets. How many distinct objects are contained in A?
Suppose that every student in a discrete mathematics class of 31 students is a freshman, a sophomore, or a junior. Prove that there could be at least 12 freshmen, at least 12 sophomores, or at least 13 juniors.
Prove that a selection of 7 numbers from the set {1, 2, . . . , 11} contains numbers m and n such that m + n = 12. (Hint: create boxes consisting of pairs that sum to 12.)
Suppose that every student in a discrete mathematics class of 25 students is a freshman, a sophomore, or a junior. a) Show that there are at least nine freshmen, at least nine sophomores, or at least nine juniors in the class. b) Show that there are either at least three freshmen, at least 19 sophomores, or at least five juniors in the class.
Let S = {0, 1}. List all the strings of length 4 over S that contain three or more 0’s.
How many strings of length 9 over the alphabet {a, b, c} are there which contain at least seven consecutive a’s? Hint: Upon writing down a few such strings you will notice that they must take one of three distinct forms (depending on where the consecutive a’s are). Then use the inclusion–exclusion principle.
By induction Show that S has 2^n subsets.
Prove that every set D ⊆ ℕ that is definable in N := (ℕ, 0, S , +, ·) is actually ∅-definable.
Suppose (X,≺) is a well-ordering. Then (X,≺) is not isomorphic to (pred(X,≺,x),≺) for any x ∈ X.
jim has precisely 150 followers. and, all 150 people who follow jim have precisely 150 followers. In other words, a follower of jim follows jim and 149 others (of course, there may be overlap among the followers). if we Let S denote the set of people that follow jim or those with whom jim has a mutual follower. What is the largest possible value of |S|?
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Suppose that 150 people celebrated their birthday in three months, where 65 people celebrated their birthday in first month, 50 celebrated their birthday in second month, 45 celebrated their birthday in third month, 30 celebrated in first and second month, 25 people celebrated in the first and third month, 24 people celebrated in the second and third month, and 11 people celebratedtheir birthdays in all three months.Find the number of people who:a) Only celebrated in first monthb) Only celebrated in second and third month but not in first monthc) Celebrated in at least one month.d) Celebrated in exactly one month.

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