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
slava [35]
3 years ago
11

Prove the following DeMorgan's laws: if LaTeX: XX, LaTeX: AA and LaTeX: BB are sets and LaTeX: \{A_i: i\in I\} {Ai:i∈I} is a fam

ily of sets, then
LaTeX: X-(A\cup B)=(X-A)\cap (X-B)

LaTeX: X-(\cup_{i\in I}A_i)=\cap_{i\in I}(X-A_i)
Mathematics
1 answer:
MariettaO [177]3 years ago
4 0
  • X-(A\cup B)=(X-A)\cap(X-B)

I'll assume the usual definition of set difference, X-A=\{x\in X,x\not\in A\}.

Let x\in X-(A\cup B). Then x\in X and x\not\in(A\cup B). If x\not\in(A\cup B), then x\not\in A and x\not\in B. This means x\in X,x\not\in A and x\in X,x\not\in B, so it follows that x\in(X-A)\cap(X-B). Hence X-(A\cup B)\subset(X-A)\cap(X-B).

Now let x\in(X-A)\cap(X-B). Then x\in X-A and x\in X-B. By definition of set difference, x\in X,x\not\in A and x\in X,x\not\in B. Since x\not A,x\not\in B, we have x\not\in(A\cup B), and so x\in X-(A\cup B). Hence (X-A)\cap(X-B)\subset X-(A\cup B).

The two sets are subsets of one another, so they must be equal.

  • X-\left(\bigcup\limits_{i\in I}A_i\right)=\bigcap\limits_{i\in I}(X-A_i)

The proof of this is the same as above, you just have to indicate that membership, of lack thereof, holds for all indices i\in I.

Proof of one direction for example:

Let x\in X-\left(\bigcup\limits_{i\in I}A_i\right). Then x\in X and x\not\in\bigcup\limits_{i\in I}A_i, which in turn means x\not\in A_i for all i\in I. This means x\in X,x\not\in A_{i_1}, and x\in X,x\not\in A_{i_2}, and so on, where \{i_1,i_2,\ldots\}\subset I, for all i\in I. This means x\in X-A_{i_1}, and x\in X-A_{i_2}, and so on, so x\in\bigcap\limits_{i\in I}(X-A_i). Hence X-\left(\bigcup\limits_{i\in I}A_i\right)\subset\bigcap\limits_{i\in I}(X-A_i).

You might be interested in
Bob has $25 to spend on the school field trip after paying $14 for the cost of the field trip and $5.25 for lunch how much will
Ganezh [65]
$5. 75 is the amount of money Bob will have left.
4 0
3 years ago
Read 2 more answers
What is the focus of the parabola?<br><br> y=−14x2−2x−2
nataly862011 [7]
First, we need to transform the equation into its standard form (x - h)²=4p(y - k).
Using completing the square method:

y = -14x² - 2x - 2
y = -14(x² + 2x/14) - 2
y = -14(x² + 2x/14 + (2/28)²) -2 + (2/28)²
y = -14(x + 1/14)² - 391/196
-1/14(y + 391/196) = (x + 1/14)²

This is a vertical parabola and its focus <span>(h, k + p) is (-1/14, -391/196 + 1/56) = (-1/14, -775/392).

Or (-0.071,-1.977).</span>
3 0
3 years ago
1.Eric earned a 12 correct points out of 15 total points on his math quiz.
Sholpan [36]

Answer:

Jamie, because 9 out of 10 is a 90 and 12 out of 15 is an 80

4 0
3 years ago
Mark the statements that are true. an angle that measures pi/3 radions also measures 30 degrees
Aleks04 [339]

Answer:

Pi/3 radians also measures 60 degrees not 30

Step-by-step explanation:

Pi/3 x 180/pi=180pi/3pi

Since both have pi they cancel out leaving 180/3, which equals 60.

8 0
3 years ago
In ABC with the right angle at vertex C, sin(b)= (26)/8. What is sin(a)?
Marat540 [252]

Answer:

We can use the Pythagorean theorem and properties of sines, cosines, and tangents to solve the triangle, that is, to find unknown parts in terms of known parts.

Step-by-step explanation:

Pythagorean theorem:   a2 + b2 = c2.

Sines:   sin A = a/c, sin B = b/c.

Cosines:   cos A = b/c, cos B = a/c.

Tangents:   tan A = a/b, tan B = b/a.

4 0
3 years ago
Other questions:
  • An elephant eats 5% of his body weight each day if he eats 300 pounds of food,how much does it weigh?
    11·1 answer
  • Maria earns $80 per day working at a gas station. Write an algebraic expression to represent the amount of money she will earn i
    12·1 answer
  • 3x + 2y = 10 2x + 3y = 15/2
    5·1 answer
  • Catherine has $54. She plans to spend more than $20 of the money for a painting canvas. The rest will go toward paints. Each tub
    8·1 answer
  • Simplify the following expression 8*(8+4-10)<br>A.16<br>B.58<br>C.112<br>D.86​
    12·1 answer
  • What is 58.65 with a 15% tip?
    14·1 answer
  • 30 POINTS FOR PEOPLE WHO ANSWER CORRECTLY!
    6·2 answers
  • The rectangular diagram shows the design for a food
    12·2 answers
  • What is the mean of the normal distribution shown below?
    10·2 answers
  • I will give 20 points
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!