Using cardinatility of sets in discrete mathematics the value of N is real numbers Let A be a collection of sets such that X\in A if and only if X\subset N and |X|=n for some n\in N Prove that |A|=|N|.

Kaycee Roche 2021-08-02 Answered
Using cardinatility of sets in discrete mathematics the value of N is real numbers
Currently using elements of discrete mathematics by Richard Hammack chapter 18
Let A be a collection of sets such that \(\displaystyle{X}\in{A}\) if and only if \(\displaystyle{X}\subset{N}\) and \(\displaystyle{\left|{X}\right|}={n}\) for some \(\displaystyle{n}\in{N}\)
Prove that \(\displaystyle{\left|{A}\right|}={\left|{N}\right|}.\)

Want to know more about Discrete math?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

un4t5o4v
Answered 2021-08-03 Author has 14220 answers
Step 1
Given that, A is collection of sets such that \(\displaystyle{x}\in{A}\) if and only if \(\displaystyle{X}\subset{N}\) and \(\displaystyle{\left|{X}\right|}={n}\) for some \(\displaystyle{n}\in{N}\)
Here, it means that A is collection of finite subset of N
Now need to show \(\displaystyle{\left|{A}\right|}={\left|{N}\right|}.\)
It is sufficient to show A is countable.
Step 2
Use induction method to show A is countable.
a: The number of subsets of N with cardinality 1 is finite. It is trivially true.
b: Assume that, the number of subsets of N with cardinality K is finite.
Step 3
c: Need to show the number of subsets of N with cardinality \(\displaystyle{K}+{1}\) is finite.
The number of subsets of N with cardinality \(\displaystyle{K}+{1}\) is exactly rwo times of number of subsets of cardinality K as each subset has two choices either \(\displaystyle{\left({K}+{1}\right)}^{{{t}{h}}}\) term belongs to the set or not.
Thus, the number of subsets of N with cardinality \(\displaystyle{K}+{1}\) is finite.
Hence, for all \(\displaystyle{n}\in{N}\) the number of subsets of N with cardinality n is finite.
Therefore, \(\displaystyle{A}={\bigcup_{{{n}={1}}}^{{\infty}}}{A}_{{{n}}}\)
Step 4
It is known that, countable union of countable set is countable.
Here, \(\displaystyle{A}_{{{n}}}\) is countable for each \(\displaystyle{n}\in{N}\)
So, A is countable.
Hence, \(\displaystyle{\left|{A}\right|}={\left|{N}\right|}\) is proved.
Not exactly what you’re looking for?
Ask My Question
2
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-03-02
Using cardinatility of sets in discrete mathematics the value of N is real numbers Currently using elements of discrete mathematics by Richard Hammack chapter 18 Let A be a collection of sets such that X in A if and only if \(X \supset N\ \text{and} |X| = n\) for some n in N. Prove that \(|A| = |N|\).
asked 2021-08-01
Discrete mathematics cardinality using Richard Hammack's Elements of Discrete mathematics chapter 18
Prove that countable union of countable sets is countable.
asked 2021-08-05
There are 60 people taking the Discrete Mathematics classthey have a distinct roll numbers from 1 to 60. Now if they pick arbitrarily 31 students.
Claim: there exists two students with roll number a and b such that b is divisible by a.
asked 2021-08-01

Let U \(= \left\{ 1,​2, 3,​ ...,2400 ​\right\}\)
Let S be the subset of the numbers in U that are multiples of 3​, and let T be the subset of U that are multiples of 7.
Since \(\displaystyle{2400}\div{3}={800}\)​, it follows that \(n(S)=n(\left\{3 \cdot 1, 3 \cdot 2, \cdots, 3 \cdot 800\right\})=800\).
​(a) Find​ n(T) using a method similar to the one that showed that \(\displaystyle{n}{\left({S}\right)}={800}\).
(b) Find \(\displaystyle{n}{\left({S}\cap{T}\right)}\).
(c) Label the number of elements in each region of a​ two-loop Venn diagram with the universe U and subsets S and T.
Questions:Find n(T) ? Find n(SnT)

asked 2021-08-05
Express the following in set-builder notation in discrete math:
a)The set A of natural numbers divisible by 3.
b)The set B of pairs (a,b) of real numbers such that a + b is an integer.
c)The open interval C = (—2,2).
d)The set D of 20 element subsets of N.
asked 2021-08-06
Let R be a relation on Z defined by (x.y) \(\displaystyle\in\) R if and only if 5(x-y)=0. Formally state what it means for R to be a symmetric relation. Is R an equivalence relation? If so, prove it discrete math and if not, explain why it is not.
asked 2021-09-16
Some of the solution sets for quadratic equations in the next sections in this chapter will contain complex numbers such as \(\displaystyle{\frac{{-{4}+\sqrt{{-{12}}}}}{{{2}}}}\ \text{ and }\ {\frac{{-{4}-\sqrt{{-{12}}}}}{{{2}}}}\). We can simplify the first number as follows. \(\displaystyle{\frac{{-{4}+\sqrt{{-{12}}}}}{{{2}}}}\)
\(\displaystyle={\frac{{-{4}+{i}\sqrt{{{12}}}}}{{{2}}}}\)
\(\displaystyle={\frac{{-{4}+{2}{i}\sqrt{{3}}}}{{{2}}}}\)
\(\displaystyle={\frac{{{2}{\left(-{2}+{i}\sqrt{{3}}\right)}}}{{{2}}}}\)
\(\displaystyle=-{2}+{i}\sqrt{{3}}\) Simplify each of the following complex numbers.
\(\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}\)

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question
...