Countability of Functions from {0,1}: Finding a 1-1 Correspondence

In summary, the conversation discusses whether the set of all functions from {0,1} to the natural numbers is countable or uncountable. One person argues that it is countable, while the other struggles with providing a 1-1 correspondence. They discuss using ordered pairs and labeling functions, but ultimately conclude that the set from the natural numbers to {0,1} is uncountable.
  • #1
kathrynag
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0

Homework Statement


Is the set of all functions from {0,1} countable or uncountable? Provide a 1-1 correspondence with a set of know cardinality.



Homework Equations





The Attempt at a Solution


I say it is countable, but my problem is I don't really know how to provide a 1-1 correspondence.
 
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  • #2


The set of all functions from {0,1} to where?
 
  • #3


oops sorry to the natural numbers. My book has another question asking the same thing but from the natural numbers to that set. I'm just trying to get the whole idea of this down.
 
  • #4


A function from the set {0,1} to the natural numbers is a set of ordered pairs {(0,m),(1,n)} where m and n are natural numbers. Do you see how to biject this with something?
 
  • #5


Not really.
f:(0, a), (1, b)
 
  • #6


If f is the function corresponding to {(0,m),(1,n)} maybe we could label it fm,n. Then each function f:{0,1}--> N is fm,n for some choices of m and n, and all fm,n are distinct
 
  • #7


So is that basically for the set going to the natural numbers. If I want the natural numbers to {0,1}, I have {(m,n)} to {0,1} and each choice is not distinct. Uncountable.
 
  • #8


Just because the choice is not distinct doesn't mean that it's uncountable
 
  • #9


It's not onto, so uncountable?
 

1. What is countability calculation?

Countability calculation is a method used in statistics and mathematics to determine the number of elements in a set or sample. It involves counting or measuring the items in a set and using various formulas and techniques to arrive at a final count.

2. How is countability calculation useful?

Countability calculation is useful in a variety of fields, including data analysis, research, and decision-making. It allows us to quantify and understand the size and characteristics of a population or sample, and make informed decisions based on this information.

3. What are some common techniques used in countability calculation?

Some common techniques used in countability calculation include counting, sampling, randomization, and statistical analysis. These techniques may vary depending on the specific context and purpose of the calculation.

4. Can countability calculation be applied in real-world situations?

Yes, countability calculation is commonly used in real-world situations. For example, it can be used in market research to estimate the size of a target market, in public health to determine the prevalence of a disease, and in quality control to measure the number of defects in a production line.

5. How accurate is countability calculation?

The accuracy of countability calculation depends on the quality of the data and the methods used. In general, the larger the sample size and the more representative it is of the population, the more accurate the calculation will be. However, there is always a degree of uncertainty and error involved in any calculation process.

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