Ask question

# 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|. # 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|.

Question
Discrete math 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 $$|A| = |N|$$.

## Answers (1) 2021-03-03
Given that, A is collection of sets such that x in A if and only if $$X \subset N and |X|=n$$ for some n in N. Here, it means that A is collection of finite subset of N. Now need to show $$|A|=|N|$$. It is sufficient to show A is countable. Use induction method to show A is countable. The number of subsets of N with cardinality 1 is finite. It is trivially true. Assume that, the number of subsets of N with cardinality K is finite. Need to show the number of subsets of N with cardinality $$K +1$$ is finite. The number of subsets of N with cardinality K +1 is exactly two times of number of subsets of cardinality K as each subset has two choices either $$(K +1)^{th}$$ term belongs to the set or not. Thus, the number of subsets of N with cardinality $$K +1$$ is finite. Hence, for all n in N the number of subsets of N with cardinality n is finite. So, A, is equal to collection of subsets of N with cardinality n . Therefore, $$A=\bigcup_{n-1}^{\infty}A_{n}$$. Tt is known that, countable union of countable set is countable. Here, A_n is countable for each n in N . So, A is countable Hence, $$|A|=|N|$$ is proved.

### Relevant Questions asked 2021-02-22
Discrete mathematics cardinality using Richard Hammack's Elements of Discrete mathematics chapter 18 Prove that countable union of countable sets is countable. asked 2020-10-21
Discrete mathematics cardinality using Richard Hammack's Elements of Discrete mathematics chapter 18 A superset of uncountable set is uncountable. (We say A is a superset of B if B sube A. ) asked 2021-03-31
A paraglider is flying horizontally at a constant speed.Assume that only two forces act on it in the vertical direction,its weight and a vertical lift force exerted on its wings by theair. The lift force has a magnitude of 1800 N.
(a) What is the magnitude and direction of the force that theparaglider exerts on the earth ?
(b)If the lift force should suddenly decrease to 1200 N, whatwould be the vertical acceleration of the glider ? For bothquestions, take the upward direction to be the + y direction. asked 2021-05-05
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X? asked 2020-12-25
Im confused on this question for Discrete Mathematics.
Let $$\displaystyle{A}_{{{2}}}$$ be the set of all multiples of 2 except for 2. Let $$\displaystyle{A}_{{{3}}}$$ be the set of all multiples of 3 except for 3. And so on, so that $$\displaystyle{A}_{{{n}}}$$ is the set of all multiples of n except for n, for any $$\displaystyle{n}\geq{2}$$. Describe (in words) the set $$\displaystyle{A}_{{{2}}}\cup{A}_{{{3}}}\cup{A}_{{{4}}}\cup\ldots$$. asked 2021-04-25
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes). asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P. asked 2021-03-02

Let g be an element of a group G. If $$|G|$$ is finite and even, show that $$g \neq 1$$ in G exists such that $$g^2 = 1$$ asked 2021-02-09
Suppose n is an integer. Using the definitions of even and odd, prove that n is odd if and only if 3n+1 is even. asked 2021-03-02

Let u,$$v_1$$ and $$v_2$$ be vectors in $$R^3$$, and let $$c_1$$ and $$c_2$$ be scalars. If u is orthogonal to both $$v_1$$ and $$v_2$$, prove that u is orthogonal to the vector $$c_1v_1+c_2v_2$$.

...