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
Gala2k [10]
3 years ago
9

Prove, using the method of bijections, that the number of subsets of [n] that consist of an odd number of elements is the same a

s the number of subsets of [n] that consist of an even number of elements g
Mathematics
1 answer:
Makovka662 [10]3 years ago
7 0

Answer:

Since, the function has been proven to be both surjective and injective, it is therefore said to be bijective and as such the question has been proved.

Step-by-step explanation:

Let A be the set of subsets of [n] that consists of an even number of elements, and let B be the set of subsets of [n] that consists of an odd number of elements.

Let's establish a bijection from A to B.

First of all,we have to establish a function that is both surjective and injective so that it is bijective.

Let the function be "f"

To prove the "f" is injective, let A1 and A2 be two subsets and consider f(A1)=f(A2)

From that, we get 2 options;

Either; A1 - {n} =f(A1) = f(A2) = A2 - {n}

Or AI u {n} =f(A1) = f(A2) = A2 u {n}

In both cases above, we can conclude that A1 = A2 and therefore, "f" is injective.

To prove that "f" is surjective, let B be an element of the range of "f" (a subset of odd size).

If B contains "n", then B−{n} is a subset of even size that maps to B under "f". Also, if B does not contain n, then B u {n} is a subset of even size that maps to B under "f".

Since everything in the image has something in the domain that maps to it, we can say that "f" is surjective.

Since, the function has been proven to be both surjective and injective, it is therefore said to be bijective and as such the question has been proved.

You might be interested in
Question 1
12345 [234]

Answer:

A). Surface area = 222 cm²

Volume = 180 cm³

B). Surface area = 375 cm²

Volume = 360 cm³

C). % increase in surface area = 67.57%

% increase in volume = 100%

Step-by-step explanation:

In the figure attached base of a prism has been given.

A). Surface area of the prism = (Perimeter of the base of the prism) × height + 2(area of the base)

Perimeter of the base = 5 + 3 + 2 + 2 + 2 + 3 + 5 + 8

                                       = 30

Area of the base = 8×5 - 2×2 = 36 cm²

Surface area of the prism = 30×5 + 2×(36)= 222 cm²

Volume of the prism = volume of the bigger prism - volume of the smaller prism cut off

                                  = 8×5×5 - 2×2×5

                                  = 200 - 20

                                  = 180 cm³

B). Surface area of the prism if it's height is 10 cm,

Surface area = 30×10 + 2×(36) = 372 cm²

Volume of the prism = 8×5×10 - 2×2×10

                                   = 400 - 40

                                   = 360 cm³                                      

C). Increase in surface area = 372 - 222 = 150 cm²

% increase in the surface area = \frac{150}{222}\times 100 = 67.57%

Increase in volume = 360 - 180 = 180 cm³

% increase in volume = \frac{180}{180}\times 100 = 100%

4 0
2 years ago
Please help i can not fail this!!
ivann1987 [24]
No solution 
i got u fam
4 0
3 years ago
Read 2 more answers
Create an equation for the following.
bixtya [17]
1/2x + 5 = 35

You use 1/2 because she sold half of her books . Then +5 because she gained 5 back and 35 because your end total should be 35.

Hope this helps ! :)

3 0
3 years ago
Someone plz help me with this
Phoenix [80]

Answer:

5y^{6}\sqrt{2}

Step-by-step explanation:

We want to  simplify:

\sqrt{50y^{12}}

We rewrite as:

\sqrt{2\times25 \times (y^{6})^2}

We split the radical sign to obtain:

\sqrt{25} \times \sqrt{(y^{6})^2} \times \sqrt{2}

Simplify the square root for the perfect squares to get:

5y^{6}\sqrt{2}

Therefore the simplified form is: 5y^{6}\sqrt{2}

8 0
2 years ago
50 POINTS !!<br><br><br> PLEASE HELP !! ILL GIVE BRAINLIEST TO THE RIGHT ANSWERS.
Artyom0805 [142]

Answer:

12

Step-by-step explanation:

Use the pythagorean theorem (a^2+b^2=c^2). A and b are the two legs of the triangle, and c is ALWAYS the hypotenuse. Plug in the values for a and c

a^2+b^2=c^2

5^2+b^2=13^2

25+b^2=169

b^2=144

b=12

6 0
2 years ago
Read 2 more answers
Other questions:
  • The equation y = x + 1 defines the relationship between x and y, where x is the input and y is the output. Which statements abou
    9·2 answers
  • Write a function rule for each table X 0,1,3,5 f(X) 0,3,9,15
    9·1 answer
  • I’m really struggling with these questions
    11·1 answer
  • Can I get help plzzz
    8·1 answer
  • I don’t understand the question above... pls give step by step explanation...
    12·2 answers
  • The points (-18,15) and (-20,15) lie on a circle with a radius of 1. Find the coordinates of the center of the circle.
    9·1 answer
  • There are 32 students in the seventh-grade class and 25 students in the eighth-grade class at OLV. What percent is the seventh-g
    15·1 answer
  • 25% of lower department store prices
    15·1 answer
  • Wat is the word form for 8.008​
    8·2 answers
  • I dont Know what to put here<br> But Anwer Pls
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!