set theory - example exercise

[qaopm]

hello everyone, this is my first post.

I have decided to refresh my math, all by myself, following some old high school books.

anyway I like to understand what I study, but with this example I have severe difficulties understanding it.

I attempted solving the example without looking at it, but I failed miserably, entering in an endless loop of wrong assumptions,

since this book seems to be written only as a textbook to be accompanied by live lessons, it lacks several explanations of what's being done, IMO.

 

"in a class of 25 students, 7 students play basket, 13 play soccer, 9 none of the above. how many students play both basket and soccer?"

the demonstration follows:

 

Universe = 25 students;
P = the 7 basket players;
C = 13 soccer players.

we have to find P intersection C.

25 - 9 = 16 (P union C) so these are students that either play basket, soccer, or both.

now here comes something I cannot understand.

(P union C) - P = C - P

(P union C) - P is straightforward, it's to find the amount of students that only play soccer.

but the equivalence, even if accurate looking at the set diagram (two intersection sets inside a Universe set), doesn't

corresponds arithmetically, since:

(P union C) - P = 9

but

C - P = 6

and even so, the book continues writing the expression as C - P.

then it continues: C - (C - P) = C intersection P

so 13 - 9 = 4 the book says and this is the end of the example.

I think there is a lot to be explained, but the book only gives semantics of union, intersection and subtraction.

these are TOTALLY different concepts from what I'm used to, like algebraic expressions or simple arithmetics.

can someone explain? I'd appreciate a lot.

 

I applied the example with bogus set content, so I know it's correct. but still I don't understand it.

P = {a; b; c;}
C = {c; d; e; f}

 

P union C = {a; b; c; d; e; f}

 

C - P = {d; e; f}

(P union C) - P = {a, b}

P intersection C = {c}

C - (C - P) = {c}

thanks in advance.

Edited May 8, 2013 by qaopm

[John]

I think there's some confusion here between the elements of each set and the cardinality (number of elements) of each set.

 

C - P is the set containing the students who are in C and are not in P (i.e. the set of students who only play soccer, which you recognize is also the result of (P U C) - P).

 

What you've done is subtract the cardinality of P from the cardinality of C, which is |C| - |P| = 13 - 7 = 6, which is different. Three of the students in P are not in C to begin with, so they don't factor into C - P. The reason subtracting the cardinality of P from the cardinality of P U C worked is that P is a subset of P U C, and thus all 7 students in P factor into the set difference.

 

Edit: I should also add that in the example at the end of your post, you have (P U C) - P = {a, b}. However, P U C = {a, b, c, d, e, f} and P = {a, b, c}, so (P U C) - P = {d, e, f}, which you can see is the same as C - P.

Edited May 9, 2013 by John

[qaopm]

thanks a lot. yes, I made a calculation mistake, I made (P union C) - P = {a; b} but it's wrong, since I subtracted C instead of P. I make a lot of calculation mistakes, without noticing at all... I don't know what to do to parse better what I make.

Edited May 9, 2013 by qaopm

[John]

As someone else who returned to studying mathematics after a fairly lengthy absence, I feel your pain. After nearly two years of being back in, my careless errors have become less frequent (being subject to having my work graded has helped), but they still happen from time to time.

 

I guess just keep working at it, and of course feel free to post here with any questions or concerns, even if it's just to get a second opinion on work you're pretty sure is correct. For at least some textbooks, you might be able to find detailed solutions to some or all exercises online, as well.

 

In any case, take care, and good luck with your studies.

[qaopm]

thanks so much =)