Let A={0,1,2,3,4}. Define a function f from A to A by f(n)=2n mod 5.
a/ Is f one-to-one?
b/ Is f onto?
Could you show me how to solve this problem, please? I have no idea what this function is.
Your help is greatly appreciated.
f(0)=2x0=0=0 mod 5
f(1)=2x1=2=2 mod 5
f(2)=2x2=4=4 mod 5
f(3)=2x3=6=1 mod 5
f(4)=2x4=8=3 mod 5,
so f(A)={0,1,2,3,4}
a/- yes
b/- yes
The function has been defined already:
f : A -> A
where
f(n) = 2n mod 5
To determine if the function is one-to-one and onto, we can enumerate the values of x and f(x) and compare with the elements of A = {0,1,2,3,4}.
x f(x)=2n mod 5
0 0 [2*0 mod 5 = 0 mod 5 = 0]
1 2 [2*1 mod 5 = 2 mod 5 = 2]
2 4
3 1 [2*3 mod 5 = 6 mod 5 = 1]
4 3
Recall the definition of "one-to-one" is
"A function f from A to B is called one-to-one (or 1-1) if whenever
f (a) = f (b) then a = b. No element of B is the image of more than one element in A."
Since no element in B (=f(x)) can be mapped from more than one element in A, the function is one-to-one.
Recall the definition of "onto".
"A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b. All elements in B are used."
Since every element of set B (=f(x)) maps to an element in A, the function is onto.
Many links are available for the definition and explanation of the two terms, you can try the following as a start, as it explains with clear examples:
http://www.regentsprep.org/Regents/math/algtrig/ATP5/OntoFunctions.htm
Thank you so much for your replies.
Could you also verify what I did on this problem please?
The set of course numbers for a collection of math courses is:
M = {0099,1111,1113,2200,2210,2220,2300,2450,2500,3500,4500,4900}
Define a relation R on M by (x,y) in R if course numbers x and y start with the same number.
a/ Verify that R is an equivalence relation
Reflexive: yes, because course number x start with the number itself.
Symmetric: yes, because if x starts with the same number as y, then y also starts with the same number as x.
Transitive: Yes, because for all courses x,y, and z, if x starts with the same number as y and y starts with
the same number as x, then x starts with the same number as z.
R is an equivalence because it is reflexive, symmetric, and transitive.
b/ Describe the distinct equivalence classes of R.
There are 5 equivalence classes:
[1] = {0099}
[2] = {1111, 1113}
[3] = {2200,2210,2220,2300,2450,2500}
[4] = {3500}
[5] = {4500,4900}
Looks all good to me!
To determine whether the function f is one-to-one and onto, we need to analyze its behavior with respect to its inputs and outputs.
a/ One-to-one (Injective):
A function is one-to-one if for every pair of distinct inputs, it produces distinct outputs.
To check if f is one-to-one, we can compare the outputs of different inputs and see if they are equal. Let's consider two distinct inputs, n₁ and n₂, where n₁ ≠ n₂.
For n₁ = 0, f(0) = 2 * 0 mod 5 = 0.
For n₂ = 1, f(1) = 2 * 1 mod 5 = 2.
Since f(n₁) ≠ f(n₂), we can see that for these distinct inputs, the outputs are different.
We can continue this comparison for all the pairs of distinct inputs and confirm that for every pair, the outputs are different. Hence, the function f is one-to-one.
b/ Onto (Surjective):
A function is onto if every element in the codomain has a corresponding element in the domain.
To check if f is onto, we need to determine if every element in the codomain (which is also A) has a corresponding element in the domain (also A). In this case, we need to check if every element in A can be obtained as an output of f.
The function f maps every number n in A to 2n mod 5. Let's consider each element of A and find its corresponding output.
For n = 0, f(0) = 2 * 0 mod 5 = 0.
For n = 1, f(1) = 2 * 1 mod 5 = 2.
For n = 2, f(2) = 2 * 2 mod 5 = 4.
For n = 3, f(3) = 2 * 3 mod 5 = 1.
For n = 4, f(4) = 2 * 4 mod 5 = 3.
As we can see, every element of A has a corresponding element in the domain. Therefore, f is onto.
In summary:
a/ The function f is one-to-one (injective).
b/ The function f is onto (surjective).
To solve similar problems, first, understand the function definition. Then, apply the definitions of one-to-one and onto to analyze the behavior of the function with respect to its inputs and outputs.