1. ## Set Theory

Hello,
I have attached the question

I'm not sure how to do a parts ii), iii), & iv)
I don't know how to express the sets they give.

2. ## Re: Set Theory

Originally Posted by iPod
Hello,
I have attached the question
I'm not sure how to do a parts ii), iii), & iv)
I don't know how to express the sets they give.
Here is a hint: $A\cup B=\{25,30,35,40,23,33\}$

3. ## Re: Set Theory

A is the set of all multiples of 5, B contains the two numbers 23 and 33, and C is the set of all prime numbers. (ii) Their union is {x: x is a multiple of 5 or x is prime or x = 33}. I did not need to include "x= 23" because 23 is prime. (iii) The complement is the set of all composite (non-prime) numbers that are NOT multiples of 5 and not equal to 33: {x: x is composite but not 33 or a multiple of 5}. (iv) should be very easy- what numbers in A or B are prime numbers?

4. ## Re: Set Theory

Ah I see, all the numbers in the sets are limited between 20 up to and including 40. So,

$A\cup B \cup C = {25, 30, 35, 40, 23, 33, 29, 31, 37, 39}$

yeah?

I think I can take it from here...

5. ## Re: Set Theory

Originally Posted by iPod
Ah I see, all the numbers in the sets are limited between 20 up to and including 40. So,

$A\cup B \cup C = \{25, 30, 35, 40, 23, 33, 29, 31, 37, {\color{red}39}\}$
yeah?
39 is not prime.

6. ## Re: Set Theory

Hello, iPod!

I have to ask: did you write out the sets?

Given: . $\begin{Bmatrix} U &=& \{x\,|\,x\in I,\:20 < x \le 40\} \\ A &=& \{x\,|\,x\text{ is a multiple of 5\}} \\ B &=& \{23,\,33\} \\ C &=& \{x\,|\,x\text{ is prime}\} \end{Bmatrix}$

We have: . $\begin{array}{ccc}U &=& \{21,22,23\,\hdots\,40\} \\ A &=& \{25,30,35,40\} \\ B &=& \{23,\,33\} \\ C &=& \{23,29,31,37\} \end{array}$

List the following sets:

. . $(i)\;B \cap C$
$B \cap C \;=\;\{23,33\} \,\cap\,\{23,29,31,37\} \;=\;\{23\}$

$(ii)\;A \cup B \cup C$
All three sets combined into one set . . .

$A \cup B \cup C \;=\;\{23,25,29,30,31,33,35,37,40\}$

$(iii)\;\overline{A \cup B \cup C}$
This is the complement of the set in part (ii).

$\overline{A \cup B \cup C)} \;=\;\{21,22,24,26,27,28,32,34,36,38,39\}$

$(iv)\;(A \cup B) \cap C$
$(A \cup B) \cap C \;=\;\bigg(\{25,30,35,40\} \,\cup \,\{23,33\}\bigg) \,\cap \,\{23,29,31,37\}$

. . . . . . . . . . $=\;\{23,25,30,33,35,40\} \cap \{23,29,31,37\}$

. . . . . . . . . . $=\;\{23\}$

. . $(i)\;(A \cup B) \cap C$
Code:
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|           o o o       o o o           |
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$(ii)\;A \cup (B \cap C)$
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|          .o o o.      o o o           |
|       o:::::::::::o           o       |
|     o:::::A:::::o:::o     B     o     |
|    o:::::::::::o:::::o           o    |
|   .::::::::::::::::::.                |
|   o:::::::::::o:::::::o           o   |
|   o:::::::::::o:o:o:o:o           o   |
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