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
Sergeu [11.5K]
3 years ago
9

Let A be a finite, non-empty subset of R. Prove that A has a maximum and a minimum. (Recall that a maximum of a set A is an uppe

r bound for the set that belongs to A, and a minimum of a set is a lower bound for the set that belongs to the set.) Hint: This result may seem so obvious that it isn't clear h One way is to use induction on the number of elements of the set.
Mathematics
1 answer:
julia-pushkina [17]3 years ago
5 0

Answer:

Let us use mathematical induction to prove the statement. So, we are going to start checking the statement for the first natural numbers.

n=1: Our set is \{x_1\}. So, obviously, x_1 is the maximum and minimum of our set. Then, the statement is true for n=1.

n=2: Our set is \{x_1,x_2\}. Necessarily, x_1 or x_1>x_2. In both cases, there is a minimum and a maximum.

Once we have our statement checked for the initial cases, we state our <em>induction hypothesis</em>:

For every finite set A of n elements there exists a maximum and a minimum.

Now, let us prove the that the above assertion is true for sets with n+1 elements.

Our set is A=\{x_1,x_2,\ldots,x_n,x_{n+1}\} and we want to find

\max\{x_1,x_2,\ldots,x_n,x_{n+1}\}.

Notice that this problem is equivalent to solve

\max\{\max\{x_1,x_2,\ldots,x_n\},x_{n+1}\},

i.e, to find the maximum among n+1 numbers, we can find first the miximum among n and then compare with the other one.

Now, using our induction hypothesis we know that there is a maximum in the set \{x_1,x_2,\ldots,x_n\}, because it has n elements. Let us write

x' =\max\{x_1,x_2,\ldots,x_n\}.

So, in order to find the maximum of A, we have to find the maximum of \A'={x',x_{n+1}\}. As we have checked at the beginning, there is a maximum in A', and it is the maximum of A.

Hence, we have completed the prove for the existence of the maximum of a set with n+1 elements. The prove for the existence of the minimum is analogue, we just need to change ‘‘maximum’’ for ‘‘minimum’’.

You might be interested in
What do the sides in a triangle add up to? I know that the inner angles of a triangle add up to 180°, but what do the sides of a
julia-pushkina [17]

Answer:

The most basic fact about triangles is that all the angles add up to a total of 180 degrees. The angle between the sides can be anything from greater than 0 to less than 180 degrees. The angles can't be 0 or 180 degrees, because the triangles would become straight lines.

Step-by-step explanation:

basic only

5 0
2 years ago
A car company is designing a car that uses 12% less steel than
Olin [163]

Answer:

180 pound

Step-by-step explanation:

<h3><u>p</u><u>e</u><u>r</u><u>c</u><u>e</u><u>n</u><u>t</u><u>a</u><u>g</u><u>e</u><u>:</u></h3>

<u>t</u><u>h</u><u>e</u><u> </u><u>v</u><u>a</u><u>l</u><u>u</u><u>e</u><u> </u><u>o</u><u>f</u><u> </u><u>1</u><u>2</u><u>%</u>

  • 12% of 1500 pounds
  • 12/100 × 1500
  • 18,000/100
  • 180

much steel will the new car model use is 180 pound

5 0
2 years ago
Do the questions that are circled also GCF stands for greatest common factor
Marizza181 [45]
I’m not sure how your class works, but usually if they are circled it means you only have to do the circled ones. GCF always stands for greatest common factor
6 0
2 years ago
Read 2 more answers
If f(x) = 3x - 2 and g(x) = 2x + 1, find (f+ g)(x).
Leto [7]
The answer would be
5x-1
3 0
3 years ago
Enter the words as an expression.<br><br> Subtract 15 from 60, then divide by 9.
Andru [333]

Answer:

60-15/9

Step-by-step explanation:

Subtract (-) 15 from 60: 60-15

Divide (/) by 9: 60-15/9

7 0
3 years ago
Read 2 more answers
Other questions:
  • 2. ** You have $10 to spend on lunch today and want both a drink and a salad. The drinks cost $1.25 each
    8·1 answer
  • An angle is a right angle if and only if it measures 90
    8·2 answers
  • What is the sum of 6 + -6 =
    14·2 answers
  • A zombie lost 3 limbs each day for 4 days. How many limbs did the zombie lose in total
    8·1 answer
  • Troy took out an 80/20 mortgage to buy a house costing $100,000. The first (80%) mortgage has an interest rate of 4.75%, The sec
    15·1 answer
  • The diameter of a circle is 14 ft. Find its area to the nearest tenth.
    12·1 answer
  • SOMEONE PLS HELP ASAP
    15·1 answer
  • Can you please ans quicklyyy
    11·1 answer
  • 12<br> 36<br> 27<br> Write<br> proportion for<br> the set of simular polygons.
    5·1 answer
  • 90°<br> Find mZR.<br> A. 30°<br> B. 60°<br> C. 120°<br> D. 180°<br> (2x)<br> to<br> P<br> 90°<br> S
    8·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!