1. ## MemberShip Tables

Hey All,

I have this question which I have given a decent stab in the dark.. ;\

It is to prove: (S \ T) U (T \ S) = (S U T) \ (S n T)

I have this so far:

Is it right, near, or somewrong? Ones highlighted in yellow im not sure about :\[IMG]file:///C:/Users/Jason/AppData/Local/Temp/moz-screenshot.png[/IMG]

2. Originally Posted by ramdrop
Hey All,

I have this question which I have given a decent stab in the dark.. ;\

It is to prove: (S \ T) U (T \ S) = (S U T) \ (S n T)

I have this so far:

Is it right, near, or somewrong? Ones highlighted in yellow im not sure about :\[IMG]file:///C:/Users/Jason/AppData/Local/Temp/moz-screenshot.png[/IMG]
I am unsure of your approach. If you draw a Venn diagram it's pretty obvious and much less work.

3. Proof:

If x in (S \ T) U (T \ S) then x in (S \ T) or x in (T \ S). If x in (S \ T) so x in S; If x in (T \ S) so x in T; Anyway x in S U T.

If x S\T so x not in T and if x in T\S so x not in S, so anyway x not in T or x not in S, therefor x not in S n T, so we get: x in (S U T) \ (S n T).

If x in (S U T) \ (S n T) then x in S U T. Therefor x in S or x in S. If x in S then x not in T . Therefor x in S\T.
Similar, if x in T\S then x in T\S. Hence: x in (S \ T) U (T \ S).

We proved that (S U T) \ (S n T) c (S \ T) U (T \ S) and (S \ T) U (T \ S) c (S U T) \ (S n T) therefor (S \ T) U (T \ S) = (S U T) \ (S n T).

4. I'm wondering whether using a Venn diagram could be considered rigorous, or whether it must be considered hand-wavy?

Or maybe this:

Let

A = S \ T
B = S n T
C = T \ S
D = S U T

A, B, C are pairwise disjoint because for example (x in S \ T) means (x in S and x not in T), which contradicts with both (x in S and x in T) and (x in T and x not in S).

A U B U C = D because (x in S) implies (x in S and x in T) or (x in S and x not in T), and likewise for (x in T).

So we take (A U B U C) \ B = D \ B and get the desired answer.

5. Okay, well let me write the problem:

Originally Posted by QUESTION
A mixed hospital ward contains 30 patients. Determine how many patients are
female who are under the age of 50 and do not suffer from a back problem
given the following information:
13 patients are at least 50 years old;
15 patients suffer from a back problem;
12 patients are male;
5 male patients have a back problem of which 2 are at least 50 years old;
5 patients are female, are at least 50 years old and do not have a back problem.
Exhibit this information in a Venn diagram. Use A to denote the set of patients
who are at least 50 years old; B to denote those with a back problem; and M
for those that are male. Let x denote the number of male patients who are at
least 50 years old but do not suffer from a back problem. (Note: the above
information is not sufficient to determine x but is sufficient to answer the
problem posed.)
For the Venn Diagram, I have..

Where Red is "Patients at least 50 years old"
Blue is Males
Green is patients with back problem
Orange is Females

Using the info above, here is where Iv put the numbers...

IS this right or?..

6. Originally Posted by ramdrop
For the Venn Diagram, I have..

Where Red is "Patients at least 50 years old"
Blue is Males
Green is patients with back problem
Orange is Females
Why is there no intersection between green and red?

7. Hello, ramdrop!

If that is a truth table, I don't understand it.

What are $\displaystyle X$ and $\displaystyle Y$?
And why are its columns only partially filled?

And exactly where is your conclusion?

Prove: .$\displaystyle (S - T) \cup (T - S) \;=\; (S \cup T) - (S \cap T)$

This can be proved with laws of logic.

$\displaystyle \begin{array}{ccc}(S\cup T) - (S \cap T) && \text{Given} \\ \\ (S \cup T) \cap \overline{(S \cap T)} && \text{d{e}f. Subtraction} \\ \\ (S \cup T) \cap (\overline S \cup \overline T) && \text{DeMorgan's Law} \\ \\ (S \cap \overline S) \cup (S \cap \overline T) \cup (T \cap \overline S) \cup (T \cap \overline T) && \text{Distributive Law} \\ \\ f \cup (S \cap \overline T) \cup (T\cap \overline S) \cup f && P \cap \overline P \:=\:f \\ \\ (S \cap \overline T) \cup (T \cup \overline S) && f \cup P \:=\:P \\ \\ (S - T) \cup (T - S) && \text{d{e}f. Subtraction} \end{array}$

8. Thank you Soroban, that really put things into perspective for me, so +1 for you

This (I hope) is the fix to the venn diagram for the question:

Originally Posted by QUESTION
A mixed hospital ward contains 30 patients. Determine how many patients are
female who are under the age of 50 and do not suffer from a back problem
given the following information:
13 patients are at least 50 years old;
15 patients suffer from a back problem;
12 patients are male;
5 male patients have a back problem of which 2 are at least 50 years old;
5 patients are female, are at least 50 years old and do not have a back problem.
Exhibit this information in a Venn diagram. Use A to denote the set of patients
who are at least 50 years old; B to denote those with a back problem; and M
for those that are male. Let x denote the number of male patients who are at
least 50 years old but do not suffer from a back problem. (Note: the above
information is not sufficient to determine x but is sufficient to answer the
problem posed.)

9. Originally Posted by ramdrop
Thank you Soroban, that really put things into perspective for me, so +1 for you

This (I hope) is the fix to the venn diagram for the question:

Honestly I'm not sure how to do the Venn diagram in this case, but reading more carefully they want you to use three sets, not four.

Here's a way just using logic, where the steps needed to answer the question are colored green

30 total
13 at least 50
=> 17 under 50

30 total
15 with back problem
=> 15 without back problem

30 total
12 male
=> 18 female

12 male
5 male with back problem
=> 7 male without back problem

15 without back problem
7 male without back problem
=> 8 female without back problem

5 male with back problem
2 male with back problem at least 50
=> 3 male with back problem under 50

8 female without back problem
5 female at least 50 without back problem
=> 3 female under 50 without back problem

10. Hello, ramdrop!

There are only three circles in the Venn diagram!
Anything outside the Male circle is a female.

A mixed hospital ward contains 30 patients.
13 are 50 or older.
15 have back problems.
12 are male.
5 are males with back problems.
2 are males with back problems and are 50 or older.
5 are female, are 50 or older and do not have back problems.

Let $\displaystyle A$ = "Older" (50 or older).
. . .$\displaystyle B$ = "Back" (back problems)
. . .$\displaystyle M$ = "Male"
. . .$\displaystyle x \:=\:\text{(Male) } \cap \text{ (Older) } \cap \sim\!\text{(Back)}$

How many patients are female, under 50 and have no back problems?

Contruct the Venn diagram and insert the known quanatities and the $\displaystyle x.$
. . It looks like this . . .

Code:
*---------------------------*
|                           |
|       *-----------*       |
|       | A         |       |
|       | 5 *-------+---*   |
|       |   |  6-x  | B |   |
|   *---+---+---*   |   |   |
|   |   | x | 2 | 3 |   |   |
|   |   *---+---+---*   |   |
|   |       |   |   4+x |   |
|   | 7-x   *---+-------*   |
|   |  M        |           |
|   *-----------*           |
|                      3    |
*---------------------------*

Add the quantities inside the three cicles:
. . $\displaystyle 5 + (-x) + (4+x) + x + 2 + 3 + (7-x) \;=\;27$

Since there are 30 patients on the ward,
. . there are 3 patients outside the three circles.

These are patients who are not male, are not 50 or older,
. . and do not have back problems.

These are exactly the patients the problem asked for!

. . $\displaystyle n(\text{female} \cap \text{under 50} \cap \text{no back problems}) \;=\;3$

11. Originally Posted by Soroban

There are only three circles in the Venn diagram!
Anything outside the Male circle is a female.

Contruct the Venn diagram and insert the known quanatities and the $\displaystyle x.$
. . It looks like this . . .

Code:
*---------------------------*
|                           |
|       *-----------*       |
|       | A         |       |
|       | 5 *-------+---*   |
|       |   |  6-x  | B |   |
|   *---+---+---*   |   |   |
|   |   | x | 2 | 3 |   |   |
|   |   *---+---+---*   |   |
|   |       |   |   4+x |   |
|   | 7-x   *---+-------*   |
|   |  M        |           |
|   *-----------*           |
|                      3    |
*---------------------------*

Thanks Soroban, just want to point out a small typo, the 3 is misplaced, should be

Code:
*---------------------------*
|                           |
|       *-----------*       |
|       | A         |       |
|       | 5 *-------+---*   |
|       |   |  6-x  | B |   |
|   *---+---+---*   |   |   |
|   |   | x | 2 |   |   |   |
|   |   *---+---+---*   |   |
|   |       | 3 |   4+x |   |
|   | 7-x   *---+-------*   |
|   |  M        |           |
|   *-----------*           |
|                      3    |
*---------------------------*

Using four sets like the OP did is only useful for when it's possible to be neither male nor female (or both, but then the diagram would look more complicated, see here)... Just for fun: assuming it's possible to be neither male nor female, but that by chance none of the 30 patients fall into this category, then you could draw this diagram:

But to be clear I do not recommend drawing the diagram this way.

12. Wow cheers guys. that really cleared it up for me - just one last question, if i was to find the largest and smallest possibility for the number of male patients under 50 without a back problem, how would i go about that?

13. Originally Posted by ramdrop
Wow cheers guys. that really cleared it up for me - just one last question, if i was to find the largest and smallest possibility for the number of male patients under 50 without a back problem, how would i go about that?
I think you can just consider simultaneous inequalities

$\displaystyle \displaystyle x \ge 0$

$\displaystyle 7-x \ge 0$

$\displaystyle 6-x \ge 0$

$\displaystyle 4+x \ge 0$

14. hmmm no idea how I would set that out :\ i lgive it a go tho

15. Originally Posted by ramdrop
hmmm no idea how I would set that out :\ i lgive it a go tho
Solve for x in each inequality; keep the strict bounds; discard the loose bounds. (They are all connected by "AND".)

For example, suppose x > -10 AND x > -4 AND x < 13 AND x < 15. It's clear that x < 15 and x > -10 give you no useful information, and you conclude that -4 < x < 13.

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