Answer to Question #244595 in Discrete Mathematics for Alina

Question

Find out if the following sets are Countable, Uncountable, Finite or if it cannot be determined. Give the reasoning behind your answer for each.

(a) Subset of a countable set

(b) integers divisible by 5 but not by 7

(c) (3, 5)

(d) A - B (A is an Uncountable set and B is a Countable set)

(e) P(C) where C is a finite set

Answer 1

First step to answer the question is to determine given types of sets

Finite is the set which contains finite number of elements

Countable is the set every element of which can be uniquely matched with natural number

Uncountable is the set which is not countable

So, technically, every finite set is also countable

(a) Subset of countable set can be both finite and countable, for example:

Let N be the set of natural numbers, and A={1,2} and B={2k, k$\in$N} both subsets of N

The set A is contains 2 elements, so it is finite, the set B is unfinite but countable by definition

(b) A={(5k, k$\in$Z)\(7k, k$\in$Z)}

Since we could match every positive integer with even positive integer, and every negative integer(with zero) with odd positive integer, the set of all numbers is also countable. Then, set A is countable by definition

(c) There is no minimal real number greater than 3, so given set could not be entirely matched with N, so it is uncountable

(d) Uncountable set minus countable set is always uncountable set(it is like removing countable amount of points from the line segment would not change the lenght covered by that segment)

(e) P(C) contains $2^{k}$ k elements where k is number of elements in set C. Since set C is finite, then k$\in$N, then $2^k\in N$, then P(C) is finite