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
pashok25 [27]
2 years ago
6

Prove each of the following statements below using one of the proof techniques and state the proof strategy you use.

Mathematics
1 answer:
pochemuha2 years ago
7 0

Answer:

See below

Step-by-step explanation:

a) Direct proof: Let m be an odd integer and n be an even integer. Then, there exist integers k,j such that m=2k+1 and n=2j. Then mn=(2k+1)(2j)=2r, where r=j(2k+1) is an integer. Thus, mn is even.

b) Proof by counterpositive: Suppose that m is not even and n is not even. Then m is odd and n is odd, that is, m=2k+1 and n=2j+1 for some integers k,j. Thus, mn=4kj+2k+2j+1=2(kj+k+j)+1=2r+1, where r=kj+k+j is an integer. Hence mn is odd, i.e, mn is not even. We have proven the counterpositive.

c) Proof by contradiction: suppose that rp is NOT irrational, then rp=m/n for some integers m,n, n≠. Since r is a non zero rational number, r=a/b for some non-zero integers a,b. Then p=rp/r=rp(b/a)=(m/n)(b/a)=mb/na. Now n,a are non zero integers, thus na is a non zero integer. Additionally, mb is an integer. Therefore p is rational which is contradicts that p is irrational. Hence np is irrational.

d) Proof by cases: We can verify this directly with all the possible orderings for a,b,c. There are six cases:

a≥b≥c, a≥c≥b, b≥a≥c, b≥c≥a, c≥b≥a, c≥a≥b

Writing the details for each one is a bit long. I will give you an example for one case: suppose that c≥b≥a then max(a, max(b,c))=max(a,c)=c. On the other hand, max(max(a, b),c)=max(b,c)=c, hence the statement is true in this case.

e) Direct proof: write a=m/n and b=p/q, with m,q integers and n,q nonnegative integers. Then ab=mp/nq. mp is an integer, and nq is a non negative integer. Hence ab is rational.

f) Direct proof. By part c), √2/n is irrational for all natural numbers n. Furthermore, a is rational, then a+√2/n is irrational. Take n large enough in such a way that b-a>√2/n (b-a>0 so it is possible). Then a+√2/n is between a and b.

g) Direct proof: write m+n=2k and n+p=2j for some integers k,j. Add these equations to get m+2n+p=2k+2j. Then m+p=2k+2j-2n=2(k+j-n)=2s for some integer s=k+j-n. Thus m+p is even.

You might be interested in
when a figure translated, reflected, or rotated, whats the same about the original figure and its image
jek_recluse [69]

Answer:

the original figure doesnt change shape, only location

Step-by-step explanation:

7 0
3 years ago
Show that cos 30 degree + sin 60 degree / 1+ sin 30 degree + cos 60 degree is equal to cos 30 degree​
Soloha48 [4]

Answer:

cos 30

Step-by-step explanation:

cos 30 + sin 60 / 1+ sin 30 + cos 60

= root3 / 2 + root3/2 / 1 + 1/2 + 1/2

= root 6/2 by 4/2

= root 6/4

= root 3/2

= cos 30

5 0
2 years ago
Wats the correct answer here
nevsk [136]
Well I think it's 10 hours and 20 minutes you just have to count the hours but subtract one hour because of lunch then add the remaining minutes.
4 0
3 years ago
Read 2 more answers
The circumference of a circle is 3pi in. What is the area of the circle?
Nezavi [6.7K]

Answer:

A = 2.25 pi in^2

Step-by-step explanation:

The circumference of a circle is 3pi

so d = 3

Area of circle = pi r^2

r = 3/2 = 1.5

A = (1.5)^2 pi

A = 2.25 pi in^2

3 0
3 years ago
Find the domain and range of the following function f(x) = 3lx+7l-2
Vesna [10]

Answer:

The Domain is all real numbers.  there are no restrictions upon x.

The Range (-2,∞) because when x= -7, -2 is the minimum amount

Step-by-step explanation:

7 0
3 years ago
Other questions:
  • TAKE A SNIPER TAKE A SHOT TAKE THESE POINTS.
    11·1 answer
  • Find the zeroes of the function f(x) = 3x + 24
    11·1 answer
  • If emma buys 7 pounds of apples and bananas and each pound of apples cost $1.75 and a pound of oranges cost$2.00, how many pound
    7·1 answer
  • Please help me solve log7 -2b-9=-6
    12·1 answer
  • Ivan and kate live 42 miles apart. ivan leaves his house at 8:00 a.m. and bikes towards kate's house at a constant speed of 12 m
    14·1 answer
  • PLS PLS ILL GIVE U BRAINLIST AND ALL THE POINT SPLS HELP ME OUT!!! HASS TO BE RIGHT ASAP PLS
    6·1 answer
  • Can someone please help? will give brainliest
    11·2 answers
  • Guinea made a scale drawing of a window she is making for her house.
    12·1 answer
  • Identify whether each phrase is an expression, equation, or inequality.
    9·2 answers
  • Area, Volume, Surface Area-
    10·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!