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Consider the set of all functions f: {0,1, … ,2014} → {0,1, … ,2014} such that f(f(i)) = i, for all 0 ≤ i ≤ 2014. Consider the following statements:
P. For each such function it must be the case that for every i, f(i) = i.
Q. For each such function it must be the case that for some i, f(i) = i.
R. Each such function must be onto.
Q.
Which one of the following is CORRECT?
Which one of the following is CORRECT?
- a)P, Q and R are true
- b)Only Q and R are true
- c)Only P and Q are true
- d)Only R is true
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider the set of all functions f: {0,1, … ,2014} → {0,1...
This kind of functions are called identity functions. We assume f(i) = k. So, f(k) = i. Now, since the values of ' i ' and ' j ' would be same for atleast some values if the domain and co - domain intersect, which is true for the given question, Q is definitely true. But this might not happen for all the values of ' i ', hence, P is not always true. Now, ' i ' ranges from 0 to 2014, so, it takes 2015 possible values. From the definition of a function, we know that for each input to the function, we have a unique output. Also, in the given question, domain and co - domain are exactly same. Therefore, the function is onto and hence, R is definitely true. Thus, the correct option is B.
Most Upvoted Answer
Consider the set of all functions f: {0,1, … ,2014} → {0,1...
To answer these questions, we need to understand the concept of a function and the set notation.
A function is a relation that maps each element from one set (called the domain) to another set (called the codomain) such that each element in the domain is associated with exactly one element in the codomain. In other words, for each input, there is exactly one output.
The set notation {0, 1, ..., n} represents a set of consecutive integers starting from 0 and ending at n. For example, {0, 1, 2} represents the set containing the integers 0, 1, and 2.
Now let's answer the questions:
1. How many functions are there from {0, 1, 2} to {0, 1}?
To determine the number of functions from a set A to a set B, we need to calculate the number of possible mappings. Since each element in the domain (set A) can be associated with any element in the codomain (set B), we have two choices for each element in the domain. Therefore, the total number of functions from {0, 1, 2} to {0, 1} is 2^3 = 8.
2. How many functions are there from {0, 1, 2, 3} to {0, 1}?
Using the same logic as above, we have two choices for each element in the domain, resulting in 2^4 = 16 possible functions.
3. How many functions are there from {0, 1, 2, 3} to {0, 1, 2}?
For each element in the domain, we have three choices as the codomain has three elements. Therefore, the total number of functions from {0, 1, 2, 3} to {0, 1, 2} is 3^4 = 81.
4. How many functions are there from {0, 1, 2, 3} to {0, 1, 2, 3}?
For each element in the domain, we have four choices as both the domain and codomain have four elements. Therefore, the total number of functions from {0, 1, 2, 3} to {0, 1, 2, 3} is 4^4 = 256.
In summary:
1. Number of functions from {0, 1, 2} to {0, 1} = 8
2. Number of functions from {0, 1, 2, 3} to {0, 1} = 16
3. Number of functions from {0, 1, 2, 3} to {0, 1, 2} = 81
4. Number of functions from {0, 1, 2, 3} to {0, 1, 2, 3} = 256
A function is a relation that maps each element from one set (called the domain) to another set (called the codomain) such that each element in the domain is associated with exactly one element in the codomain. In other words, for each input, there is exactly one output.
The set notation {0, 1, ..., n} represents a set of consecutive integers starting from 0 and ending at n. For example, {0, 1, 2} represents the set containing the integers 0, 1, and 2.
Now let's answer the questions:
1. How many functions are there from {0, 1, 2} to {0, 1}?
To determine the number of functions from a set A to a set B, we need to calculate the number of possible mappings. Since each element in the domain (set A) can be associated with any element in the codomain (set B), we have two choices for each element in the domain. Therefore, the total number of functions from {0, 1, 2} to {0, 1} is 2^3 = 8.
2. How many functions are there from {0, 1, 2, 3} to {0, 1}?
Using the same logic as above, we have two choices for each element in the domain, resulting in 2^4 = 16 possible functions.
3. How many functions are there from {0, 1, 2, 3} to {0, 1, 2}?
For each element in the domain, we have three choices as the codomain has three elements. Therefore, the total number of functions from {0, 1, 2, 3} to {0, 1, 2} is 3^4 = 81.
4. How many functions are there from {0, 1, 2, 3} to {0, 1, 2, 3}?
For each element in the domain, we have four choices as both the domain and codomain have four elements. Therefore, the total number of functions from {0, 1, 2, 3} to {0, 1, 2, 3} is 4^4 = 256.
In summary:
1. Number of functions from {0, 1, 2} to {0, 1} = 8
2. Number of functions from {0, 1, 2, 3} to {0, 1} = 16
3. Number of functions from {0, 1, 2, 3} to {0, 1, 2} = 81
4. Number of functions from {0, 1, 2, 3} to {0, 1, 2, 3} = 256
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Consider the set of all functions f: {0,1, … ,2014} → {0,1, … ,2014} such that f(f(i)) = i, for all 0 ≤ i ≤ 2014. Consider the following statements:P. For each such function it must be the case that for every i, f(i) = i.Q. For each such function it must be the case that for some i, f(i) = i.R. Each such function must be onto.Q.Which one of the following is CORRECT?a)P, Q and R are trueb)Only Q and R are truec)Only P and Q are trued)Only R is trueCorrect answer is option 'B'. Can you explain this answer?
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