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
GalinKa [24]
3 years ago
13

Let A, B, C and D be sets. Prove that A \ B and C \ D are disjoint if and only if A ∩ C ⊆ B ∪ D

Mathematics
1 answer:
ANEK [815]3 years ago
7 0

Step-by-step explanation:

We have to prove both implications of the affirmation.

1) Let's assume that A \ B and C \ D are disjoint, we have to prove that A ∩ C ⊆ B ∪ D.

We'll prove it by reducing to absurd.

Let's suppose that A ∩ C ⊄ B ∪ D. That means that there is an element x that belongs to A ∩ C but not to B ∪ D.

As x belongs to A ∩ C, x ∈ A and x ∈ C.

As x doesn't belong to B ∪ D, x ∉ B and x ∉ D.

With this, we can say that x ∈ A \ B and x ∈ C \ D.

Therefore, x ∈ (A \ B) ∩ (C \ D), absurd!

It's absurd because we were assuming that A \ B and C \ D were disjoint, therefore their intersection must be empty.

The absurd came from assuming that A ∩ C ⊄ B ∪ D.

That proves that A ∩ C ⊆ B ∪ D.

2) Let's assume that A ∩ C ⊆ B ∪ D, we have to prove that A \ B and C \ D are disjoint (i.e.  A \ B ∩ C \ D is empty)

We'll prove it again by reducing to absurd.

Let's suppose that  A \ B ∩ C \ D is not empty. That means there is an element x that belongs to  A \ B ∩ C \ D. Therefore, x ∈ A \ B and x ∈ C \ D.

As x ∈ A \ B, x belongs to A but x doesn't belong to B.  

As x ∈ C \ D, x belongs to C but x doesn't belong to D.

With this, we can say that x ∈ A ∩ C and x ∉ B ∪ D.

So, there is an element that belongs to A ∩ C but not to B∪D, absurd!

It's absurd because we were assuming that A ∩ C ⊆ B ∪ D, therefore every element of A ∩ C must belong to B ∪ D.

The absurd came from assuming that A \ B ∩ C \ D is not empty.

That proves that A \ B ∩ C \ D is empty, i.e. A \ B and C \ D are disjoint.

You might be interested in
I have 5 digits. My 4s are worth 4 (10000) and 4 ×10. My 3 is worth 3. The other is worth one tenth as much. My other digit is a
likoan [24]
The number is 40243 :)
hope this helps
4 0
3 years ago
K=am+3mx solve for m
klemol [59]
M is equal to the value m
7 0
3 years ago
Guys, Can you please help me with these questions
MakcuM [25]

Answer:

a) w^{13} x^{5} y^{6}

b) \frac{x}{3y^{6} }

Step-by-step explanation:

a) (w^{2} xy^{3} )^{2}(w^{3}x )^{3}

1. Distribute the second power (2) outside the first pair of parenthesis:

(w^{2(2)} x^{2} y^{3(2)} )

= w^{4} x^{2} y^{6} (w^{3}x )^{3}

2. Distribute the third power (3) outside the second pair of parenthesis:

(w^{3(3)} x^{3} )

= w^{4} x^{2} y^{6} w^{9} x^{3}

3. Combine like terms:

w^{13} x^{5} y^{6}

--------------------------------------------

b) \frac{2x^{2} y^{5} }{6xy^{11} }

1. Factor the number 6 (= 2 · 3):

\frac{2x^{2} y^{5} }{2(3)xy^{11} }

2. Cancel the common factor (2):

\frac{x^{2} y^{5} }{3xy^{11} }

3. Cancel out xy^{5} in the numerator an denominator:

\frac{x}{3y^{6} }

hope this helps!

4 0
2 years ago
Solve the problems using a system of equations.
evablogger [386]

step-by-step explanation:1true 22

7 0
3 years ago
Read 2 more answers
Can some one please help<br><br> Simplify 4(11-6)+2
Nata [24]

Answer: 22

Step-by-step explanation:

11 - 6 = 5

4 x 5 = 20

20 + 2 = 22

' pls can i have brainliest? i need it for my goal'

4 0
2 years ago
Read 2 more answers
Other questions:
  • If f(x) = |x| and g(x) = |x| − 4, which transformation is applied to f(x) to get g(x)
    9·1 answer
  • Find two consecutive odd integers such that their product is 71 more than 3 times their sum.
    14·1 answer
  • Create an expression, to equal 13, using any grouping of addition, subtraction, multiplication, and division. You must use all t
    13·1 answer
  • 50 PTS PLUS I WILL MARK BRAINLIEST
    13·2 answers
  • Joe had $2. He found a quarter but he lost a dollar in the vending machine. How much money does he have now
    8·2 answers
  • HELPP Find the value of x. If your answer is not an integer, leave it in simplest radical form. The diagram is not drawn to scal
    13·1 answer
  • Help ASAP WILL GIVE BRAINLY!!<br><br> Need help on all answers/ show work
    8·2 answers
  • 12-2²·2=?<br><br> Brainliest
    5·2 answers
  • I need help im so confused and its due in 20 minutes
    6·1 answer
  • Help me with my homework
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!