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]
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).
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.
Okay, well let me write the problem:
For the Venn Diagram, I have..Originally Posted by QUESTION
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?..
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
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 = "Older" (50 or older).
. . . = "Back" (back problems)
. . . = "Male"
. . .
How many patients are female, under 50 and have no back problems?
Contruct the Venn diagram and insert the known quanatities and the
. . 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:
. .
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!
. .
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.
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.