1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Hatshy [7]
3 years ago
10

Prove that composition of injective functions is injective. prove that composition of surjective functions is surjective.

Mathematics
1 answer:
pantera1 [17]3 years ago
6 0
Proof 1: Proving the composition of injective functions is itself an injective function.

f(x) is injective
f(a) = f(b) <--> a = b

g(x) is injective
g(a) = g(b) <--> a = b

prove f(g(x)) is injective
f(g(a)) = f(g(b))
f(a) = f(b) ... since g(x) is injective
a = b
So that proves if f(g(a)) = f(g(b)), then a = b

Now go the other direction
a = b
g(a) = g(b) 
f(g(a)) = f(g(b))
So that wraps up the sub-proof of "f(g(a)) = f(g(b)) iff a = b"

-------------------------

proving g(f(x)) is injective is going to follow a similar path of steps

------------------------------------------
------------------------------------------

Proof 2: Proving the composition of surjective functions is itself a surjective function.

f(x) is surjective
The range of f(x) is the set of all real numbers

g(x) is surjective
The range of g(x) is the set of all real numbers

f(g(x)) is also surjective because of property 1 listed above. The range of f(x) is the set of all real numbers. The fact that g(x) represents all reals as well means that we touch all of the inputs needed to generate all the outputs for f(x).

-------------------------

Proving g(f(x)) to be surjective follows a similar path.


You might be interested in
How can i get my answer for 5+(-34)
julia-pushkina [17]

Hello there.

Answer: ⇒⇒⇒⇒-29

Step-by-step explanation:

Remove parenthesis.

5-34

Subtract the numbers.

5-34=-29

Hope this helps!

Thank you for posting your question at here on brainly.

-Charlie

4 0
3 years ago
Suppose there are 21 students in your class if the draws two names at random
Gala2k [10]

Answer:

2/21 is the most logical answer

Step-by-step explanation:

4 0
3 years ago
Find the values of x and y
pantera1 [17]

Answer:

x = 15°

y = 59°

Step-by-step explanation:

Angles at a right angle add up to 90°

41 + y = 90

y = 90 - 41

y = 59°

Angles in a triangle add up to 180°

41 + (2x - 9) + 7x + 13 = 180

41 + 13 - 9 + 2x + 7x = 180

45 + 9x = 180

9x = 180 - 45

9x = 135

x = 135/9

x = 15°

6 0
4 years ago
Read 2 more answers
2/5m - 4/5 - 3/5m please skrelp
netineya [11]
2/5m - 4/5 - 3/5m...combine like terms
-1/5m - 4/5
7 0
3 years ago
Read 2 more answers
An item is regularly priced at $35. It is on sale for 20% off the regular price. How much (in dollars) is discounted from the re
Lisa [10]

Answer:

7

Step-by-step explanation:

you multiply 35 by .20

4 0
3 years ago
Read 2 more answers
Other questions:
  • Find p((5 on green die and 3 on red die) or (3 on green die and 5 on red die)). (enter your answer as a fraction.)
    14·1 answer
  • Please choose an answer below.<br> A) 3.4 <br> B) 36.0 <br> C) 38.6 <br> D) 41.2
    6·1 answer
  • 16/40 as a simplified fraction then a percentage WITH working out
    9·2 answers
  • Find the value of 7v-6 given that -8v-9=7 Simplify your answer as much as possible 7v-6=
    13·2 answers
  • Ellen is making jewelry sets that contain a bracelet and a pair of earrings. Each bracelet uses 3 times as many beads as one ear
    10·1 answer
  • PLA HELP ME ASAP FOR 12 and 13 (SHOW WORK!!!!!) + LOTS OF POINts!
    6·1 answer
  • A fishing boat was out to sea for 6 months and traveled a total of 8,578 miles. In the first month, the boat traveled 659 miles.
    10·2 answers
  • Please help, need an answer ASAP!!!
    9·1 answer
  • 55% of what number is 1100? Which of the following proportions could be used to
    10·1 answer
  • Simplify the following expression (87x + 63y) - (6x + 32y)
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!