Set Theory

**iPod** · July 22nd 2012, 08:42 AM

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.

**Plato** · July 22nd 2012, 08:48 AM

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: $A\cup B=\{25,30,35,40,23,33\}$

**HallsofIvy** · July 22nd 2012, 08:52 AM

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?

**iPod** · July 22nd 2012, 09:49 AM

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...

**Plato** · July 22nd 2012, 10:10 AM

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.

**Soroban** · July 22nd 2012, 12:35 PM

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).

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

$(iv)\;(A \cup B) \cap C$

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

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

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

Shade the following regions.

. . $(i)\;(A \cup B) \cap C$

Code:

      * - - - - - - - - - - - - - - - - - - - * 
      |                                       |
      |           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   |
      |   o         o o:::::::o o         o   |
      |           o:::::::::::::::o           | 
      |    o     o:::::o:::::o:::::o     o    |
      |     o    :::::::o:::o:::::::    o     | 
      |       o o:::::::::o:::::::::o o       | 
      |         o:o:o:o       o:o:o:o         | 
      |         o                   o         | 
      |                                       | 
      |          o                 o          | 
      |           o       C       o           | 
      |             o           o             | 
      |                 o o o                 | 
      |                                       | 
      * - - - - - - - - - - - - - - - - - - - *

$(ii)\;A \cup (B \cap C)$

Code:

      * - - - - - - - - - - - - - - - - - - - * 
      |                                       |
      |          .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   |
      |   o:::::::::o:o:::::::o o         o   |
      |   ::::::::o:::::::::::::::o           | 
      |    o:::::o:::::o:::::o:::::o     o    |
      |     o:::::::::::o:::o:::::::.   o     | 
      |       o:o:::::::::o:::::::::o o       | 
      |         o o o o       o:o:o:o         | 
      |         o                '''o         | 
      |                                       | 
      |          o-                o          | 
      |           o       C       o           | 
      |             o           o             | 
      |                 o o o                 | 
      |                                       | 
      * - - - - - - - - - - - - - - - - - - - *

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