Mathematics, 08.02.2022 03:20 masonbitterman2604

Prove that A n (B u C) = (A n B) u ( A n C)

Answers: 2

Show answers

Answers

Answer from: BakerElsie02

Let x be inside set A n (B u C)

We can write this as $x \in A \cap(B \cup C)$

This means x is in set A and also in set B u C. It may be in B, or in C, or both B and C at the same time.

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

Let's assume x is in set B but not in set C.

If so, then x is definitely in A n B and not in A n C. Overall, x is in (A n B) u ( A n C)

This leads us to the fact that A n (B u C) is a subset of (A n B) u ( A n C)

Symbolically we would write $A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$

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

Now let's assume that x is in set C but not in set B

We follow the same basic steps as the previous section. We'll conclude with the same symbolic notation written. There's not much different here other than x is in set A n C rather than A n B

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

The previous two sections fully confirm that $A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$ is true.

To prove $A \cap (B \cup C) \supseteq (A \cap B) \cup (A \cap C)$ is true, we will use the same idea.

Let y be a member of the set (A n B) u ( A n C)

This means y is either in A n B or it is in A n C

If y is in A n B, then it's certainly in A and B. It doesn't matter if it's in C or not. We have enough to show that y is a member of A n (B u C)

Or if y is in A n C, then it's in A and C simultaneously. This places element y in the set A n (B u C) overall.

Therefore, we have proven that $A \cap (B \cup C) \supseteq (A \cap B) \cup (A \cap C)$ is true. If something is in the right-side set, then it's also in the left-side set as well.

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

To wrap things up, we have proven these two claims

$A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)\\\\A \cap (B \cup C) \supseteq (A \cap B) \cup (A \cap C)$

In other words, we have shown the left and right sides are subsets of each other. This is sufficient to conclude that the two sets are equal.

Therefore, we have shown that

$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

is a true claim.

For more information, check out the distributive property dealing with set mathematics.

Answer from: Quest

idk but i found this "the graph of y = f(x-2) is the graph of y = f(x) translated two units rightward.

so, since the point (9,2) lies on the graph of y = f(x), the point (11,2) lies on the graph of y = f(x-2)."

step-by-step explanation:

link is

Answer from: Quest

he will have 300 because he is really broke and 300 is in his imangine

step-by-step explanation: he ehe heh step by the thy

Another question on Mathematics

Mathematics, 21.06.2019 20:30

Me n o w. i need t h i s. 10 points

Answers: 1

Answer

Mathematics, 21.06.2019 21:40

The number of chocolate chips in a bag of chocolate chip cookies is approximately normally distributed with a mean of 12611261 chips and a standard deviation of 118118 chips. (a) determine the 2626th percentile for the number of chocolate chips in a bag. (b) determine the number of chocolate chips in a bag that make up the middle 9696% of bags. (c) what is the interquartile range of the number of chocolate chips in a bag of chocolate chip cookies?

Answers: 2

Answer

Mathematics, 21.06.2019 22:10

Gravel is being dumped from a conveyor belt at a rate of 25 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. how fast is the height of the pile increasing when the pile is 14 ft high? (round your answer to two decimal places.) ft/min

Answers: 3

Answer

Mathematics, 21.06.2019 23:00

Eden has a part time job. she is paid ? 7.20 an hour the week she worked 18 and a half hours how much is she paid

Answers: 1

Answer

You know the right answer?

Prove that A n (B u C) = (A n B) u ( A n C)...

Questions

Mathematics, 16.09.2020 02:01

What is the answer to 3x10¹=...

Mathematics, 16.09.2020 02:01