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
3/4 of the weight of a bunch of grapes is equivalent to 3/5 of the weight of a papaya. If the grapes way (X +28) pounds what is
Margaret [11]
Let p = weight of papaya and g = weight of grapes.

Then (3/4)g = (3/5)p.  Since the weight of grapes is x + 28, 

(3/4)(x + 28) = (3/5)p.  We must solve for x.  To do this, mult. both sides by (5/3):

(5/3)(3/4)(x+28) = (5/3)(3/5)p

Then p = (15/9)(x+28), or (after reduction), p = (5/3)(x+28).


4 0
3 years ago
The table shows your test results for math class. What test score do you need on the last exam to earn 90% of the total points?
Nady [450]

Answer:

262.6/300% for the first 3 test if you were to add another test it would be 400% total and you need 360% of test grades to have an average of 90% and so 360 - 262.6 = 97.4%

Step-by-step explanation:

4 0
3 years ago
What is the measure of θ in radians?<br> Enter an exact expression.
Sonja [21]

Answer:

The answer is \frac{7\pi }{6}

Step-by-step explanation:

Just did this problem

4 0
3 years ago
Solve for m.<br> 2=​​m/2−7<br> two step equation
Svetradugi [14.3K]

2=m/2−7

Simplify both sides of the equation.

2=m/2−7

2=1/2m+−7

2=1/2m−7

Flip the equation.

1/2m−7=2

Add 7 to both sides.

1/2m−7+7=2+7

1/2m=9

Multiply both sides by 2.

2*(1/2m)=(2)*(9)

m=18

4 0
3 years ago
Type the equation that shows the pattern among this set of ordered pairs.
stich3 [128]
First find the slope between any two points:

\sf Slope=\frac{y_2-y_1}{x_2-x_1}

Where \sf (x_1,y_1),(x_2,y_2) are the two points

So calculating the slope:

\sf Slope=\frac{5-0}{2-1}=\frac{5}{1}=5

So the slope is \boxed{5}

And our equation will be in the format of \sf y=mx+b
where \sf m~ is~ the ~slope and \sf b ~is~the~y-intercept

So, now we have half of the equation:

\sf y= 5x+b

Now to calculate b, we can plug in a point \sf (x,y) and solve for b.

So

\sf y= 5x+b

Lets use the point
\sf (1,0) which ~is~in~the~form ~of ~(x,y)

So:
\sf 0=5(1)+b

And then:
\sf b=-5

So our final equation is \sf \boxed{y=5x-5}
6 0
3 years ago
Other questions:
  • I need help with his math maze please
    11·1 answer
  • (-12) + 22 +(-5) ×100=
    14·2 answers
  • If p= the game systems original price in dollars, which algebraic expression represents the reduced price
    5·1 answer
  • Milena's take-home pay is $1200 a month. She spends 12% of her take-home pay on her cable bill. How much is Milena's monthly cab
    11·2 answers
  • clayton uses 34 of a bottle of hot sauce for every batch of salsa he makes. yesterday, he used 214 bottles of hot sauce. how man
    15·1 answer
  • Are the triangles above similar?
    9·2 answers
  • Can anyone help with this:
    13·1 answer
  • Who is correct alex or jo
    6·1 answer
  • Whats This Questions Answer Answer It Cuz I Have math Exam After 3Days
    9·1 answer
  • System of equations- does this have one solution, many solutions or no solution
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!