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Anna11 [10]
3 years ago
12

Evaluate f plz easy 20 pts

Mathematics
1 answer:
noname [10]3 years ago
3 0

Answer:

the 2nd option

Step-by-step explanation:

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3. Let A, B, C be sets and let ????: ???? → ???? and ????: ???? → ????be two functions. Prove or find a counterexample to each o
Fiesta28 [93]

Answer / Explanation

The question is incomplete. It can be found in search engines. However, kindly find the complete question below.

Question

(1) Give an example of functions f : A −→ B and g : B −→ C such that g ◦ f is injective but g is not  injective.

(2) Suppose that f : A −→ B and g : B −→ C are functions and that g ◦ f is surjective. Is it true  that f must be surjective? Is it true that g must be surjective? Justify your answers with either a  counterexample or a proof

Answer

(1) There are lots of correct answers. You can set A = {1}, B = {2, 3} and C = {4}. Then define f : A −→ B by f(1) = 2 and g : B −→ C by g(2) = 4 and g(3) = 4. Then g is not  injective (since both 2, 3 7→ 4) but g ◦ f is injective.  Here’s another correct answer using more familiar functions.

Let f : R≥0 −→ R be given by f(x) = √

x. Let g : R −→ R be given by g(x) = x , 2  . Then g is not  injective (since g(1) = g(−1)) but g ◦ f : R≥0 −→ R is injective since it sends x 7→ x.

NOTE: Lots of groups did some variant of the second example. I took off points if they didn’t  specify the domain and codomain though. Note that the codomain of f must equal the domain of

g for g ◦ f to make sense.

(2) Answer

Solution: There are two questions in this problem.

Must f be surjective? The answer is no. Indeed, let A = {1}, B = {2, 3} and C = {4}.  Then define f : A −→ B by f(1) = 2 and g : B −→ C by g(2) = 4 and g(3) = 4. We see that  g ◦ f : {1} −→ {4} is surjective (since 1 7→ 4) but f is certainly not surjective.  Must g be surjective? The answer is yes, here’s the proof. Suppose that c ∈ C is arbitrary (we  must find b ∈ B so that g(b) = c, at which point we will be done). Since g ◦ f is surjective, for the  c we have already fixed, there exists some a ∈ A such that c = (g ◦ f)(a) = g(f(a)). Let b := f(a).

Then g(b) = g(f(a)) = c and we have found our desired b.  Remark: It is good to compare the answer to this problem to the answer to the two problems

on the previous page.  The part of this problem most groups had the most issue with was the second. Everyone should  be comfortable with carefully proving a function is surjective by the time we get to the midterm.

3 0
3 years ago
Find the Arithmetic mean<br> between 6 and 12​
mafiozo [28]

Step-by-step explanation:

Arithmetic mean

A.M. =  \frac{6 + 12}{2}  =  \frac{18}{2}  = 9 \\

6 0
3 years ago
Read 2 more answers
Indicate whether the following variables are qualitative (QL) or quantitative (QU): a. Favorite color ____ b. Current Occupation
Whitepunk [10]

Answer:

a. Qualitative.

b. Qualitative.

c. Quantitative.

d. Quantitative.

e. Qualitative.

Step-by-step explanation:

a.

Favorite color can be classified into different categories such as blue,orange, white etc, so, favorite color is a qualitative variable.

b.

Current Occupation can be classified into different categories such as teacher, lawyer and doctor etc, so, current occupation is a qualitative variable.

c.

Number of extra credit problems solved can be meaningfully presented as numerical figures so, number of extra credit problems solved  is a quantitative variable.

d.

Number of students in class can be meaningfully presented as numerical figures so, number of of students in our class is a quantitative variable.

e.

Exam Grades can be classified into different categories such as A, B and C etc, so, exam grades is a qualitative variable.

4 0
3 years ago
What is the value of b?
Natalija [7]
b=180-90-51=39
8 0
3 years ago
How do you find the value of x^2 + 8 when x = 5
Ivahew [28]

Answer:

Step-by-step explanation:

when x = 5

x² + 8

= (5)² + 8

= 25 + 8

= 33

8 0
2 years ago
Read 2 more answers
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