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]
3 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:
pochemuha3 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
The diameter of each wheel of a bicycle is 28 inches. If you are traveling at a speed of 15 miles per hour on this? bicycle, thr
Rudik [331]

Answer:

180.07 rev / min

Step-by-step explanation:

Diameter (D) of each wheel =28 inches

Circumference (C) =  π x 28 = 28π inches

Conversions:

1 mile = 63,360 inches

1 hour = 60 min

Hence 15 miles / hour

= (15)(63,360) inches / hour

= (15)(63,360) / 60  inches / min

= 15,840 inches per min

Number of revolutions per min = Number of inches per min ÷ circumference in inches

=15,840 inches/min ÷ 28π inches

= 180.07 revolutions per min

7 0
3 years ago
A researcher wondered whether drivers treat bicycle riders differently when they wear helmets. He rigged his bicycle with an ult
andrew-mc [135]

Answer:

In this case study we have following groups:

1. Population: It had all the cars which passed the bicyclist. Out of which 3000 cars were taken.

2. Who:  Each instance of a car passing a rider, means the drivers passing by.

That is the 3000 cars which passed the researcher on his bicycle.

3. What: The distance at which cars pass the bicycle rider. The distance the drivers stayed away from his bike.

3 0
3 years ago
X) 7.A.47
vodomira [7]

Answer:

You will gain $27

Step-by-step explanation:

Just multiply $63 times 0.42857142857

8 0
3 years ago
W + 1 = d and d = 3w solve by substitution
STALIN [3.7K]

Answer:

Step-by-step explanation:

Given

w + 1 = d...........equation 1

d = 3w ...............equation 2

Now putting d = 3w in equation 1

w + 1 = 3w

3w - w = 1

2w = 1

w = 1 / 2

putting w = 1 / 2 in equation 2

d = 1 / 2 * 3

 = 3/2

Hope it helps :)

8 0
3 years ago
Read 2 more answers
Subtract u from 7, then divide v by the result
KatRina [158]
<span>Subtract u from 7, then divide v by the result</span>
v : (7 - u)
4 0
3 years ago
Other questions:
  • Walter buys a bus pass for $30. Everytime he rides the bus, money is deducted from the value of the pass. He rode 12 times and $
    11·1 answer
  • PLZZZZZZZZZ HELP!!!!!!!!!!
    12·1 answer
  • Express the numbers as a product of its prime factors . Give ur answer in power form
    8·1 answer
  • Im confused how many solutions are parellel lines and how many solutions are intersecting lines
    13·1 answer
  • Simplify
    5·2 answers
  • Which is equivalents to 9/8​
    10·2 answers
  • Find the measure of each numbered angle.
    15·2 answers
  • Answer<br> a) 4x4x4x5x5 using index notation <br><br> b) 9x7x9x9x7x9 using index notation
    15·2 answers
  • 9.
    6·1 answer
  • Find the location of G, the midpoint of FH.
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!