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
liubo4ka [24]
3 years ago
12

In each part, decide if the statement is true. If it is true, prove it. If it is not true, give an explicit counterexample and d

emonstrate that your counterexample really is one.
(a) For any sets A, B and C, A ∩ B ⊆ C if and only if either A ⊆ C or B ⊆ C.

(b) For any sets A, B and C, A ⊆ B ∩ C if and only if both A ⊆ B and A ⊆ C
Mathematics
1 answer:
svet-max [94.6K]3 years ago
4 0

Answer: Hello!

ok, remember that "if and only if" implies that you need to prove the statement in both ways, this is represented with the ⇔ usually.

a) For any sets A, B and C, A ∩ B ⊆ C if and only if either A ⊆ C or B ⊆ C.

In this type of problems, i  find very useful start looking for some counterexample.

In this case, suppose that A = {1,2,3,4,5} , B = {3,4,5,6,7} and C = {3,4,5,6}

then is easy to see that A ⊄ C and B ⊄C.

And A∩B = {3,4,5}

then A∩B ⊂ C

then the statement is false (because one of the ways is false, remember that this is an "if and only if" statement)

b) For any sets A, B and C, A ⊆ B ∩ C if and only if both A ⊆ B and A ⊆ C.

the first way is true; because if A ⊆ B ∩ C. then all the elements of A are in the intersection of B and C (which are common elements for B and C) and then all the elements of A are in the set B and in the set C, and this means that A ⊆ B and A ⊆ C.

But let's see the other way now, suppose that A ⊆ B and A ⊆ C, now we want to know if A ⊆ B ∩ C.

if A ⊆ B and A ⊆ C, means that all the elements of A are in B, and all the elements of A are in C, then all the elements of A are common elements between B and C, this means that B ∩ C is at least equal to A (at least, because we know that all the elements of A are common elements between B and C, but there could be more common elements that don belong to A)

then A ⊆ B ∩ C.

You might be interested in
Community Gym charges a $50 membership fee and a $60 monthly fee. Workout Gym charges a $180 membership fee and a $50 monthly fe
Vladimir79 [104]

Answer:

After 13 months the amount paid to both will be $830.

Step-by-step explanation:

60x13+50=830

50x13+180=830

4 0
2 years ago
A recipe for trail mix uses 7 ounces of almonds with 5 ounces of raisins. How many ounces of almonds would be in a one pound bag
Natasha2012 [34]

8 and 3/4 ounces of almonds.

3 0
3 years ago
Read 2 more answers
If f(x)=3x+2 what is the equation for f^-1(x)
Elden [556K]

Answer:

  f^-1(x) = (x -2)/3

Step-by-step explanation:

To find the inverse function, swap x and y, then solve for y.

  y = 3x +2 . . . . given function

  x = 3y +2 . . . . inverse relation

  x -2 = 3y . . . . subtract 2

  (x -2)/3 = y . . . divide by 3

f^-1(x) = (x -2)/3 . . . . in functional form

4 0
3 years ago
Help pls first correct answer gets brainleist ​
nevsk [136]

Answer:

18

Step-by-step explanation:

a^2 + b^2 = c^2

in this case 24 is a and 30 is c

24^2 = 576

30^2 = 900

900 - 576 = 324

the square root of 324 is 18

4 0
2 years ago
Two positive consecutive odd numbers of 143
NNADVOKAT [17]
You can't have two odd numbers equal an odd number(143).
5 0
3 years ago
Read 2 more answers
Other questions:
  • Write an expression for the model. Find the sum.
    8·1 answer
  • .. an. v. . it V V is going on afield trip to Austin to of Texas History and lo see an I MAX Ruiz day given the number of hours
    14·1 answer
  • HEY CAN ANYONE PLS HELP ME OUT IN DIS PLS!!!
    12·2 answers
  • Which choice could be the equation of a line parallel to the line represented by this equation?
    14·1 answer
  • Jen and Sarah go to lunch at the green grill. Their meals total $28.00. The tax is 6%. What is the total cost including tax?
    10·1 answer
  • How do you solve the problem attached?
    11·2 answers
  • Find two positive numbers if their ratio is 8:5 and the difference is 150.
    6·1 answer
  • A baseball “diamond” is a square with a side length of 90 feet. How far is the throw from
    9·1 answer
  • If a,b,c,d are in continuous proportion then prove that (c+a) (c+d) = (b+c) (b+d)​
    5·1 answer
  • Evaluate the expression a•b for a =24 and b=8
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!