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hello everyone, this is more like a simple curiosity, yesterday night I took some pictures of the stars with my camera. the first picture was aimed at Capella and the field of view was diagonal, longitudinal and the exposition was 20 seconds, everything went fine. for the second and the third picture I wanted to grab a bigger slice of the sky, so I just put the camera on the ground facing upwards, with the focus to infinity, so the field of view was completely vertical. for the first attempt I set the shutter to 30 seconds and the stars left a trail, for the second attempt I set the shutter 15 seconds and the stars left half the trail relative to the first attempt. of course I immediately blamed the terrestrial rotation (at the first attempt) but what puzzles me is that the diagonal-longitudinal shot was perfectly still. what happened? thanks in advance.

OK solved! U = 65 C = 5 A intersect B - C = 20 complementary of A union B = 3 A = 50 A - B = ? B - A = ? A - B = A - (A intersect B) + C = 50 - 20 + 5 = 25 A union B = 65 - 3 = 62 A union B - A = B - A = 62 - 50 = 12 and that's it! =) I was making the wrong diagram, since C is subset of A intersect B.

what I am a bit more sure is this calculation: S1 union S2 union S3 - S1 union S3 = S2 - (S1 union S3) = 62-55 = 7 and then I have to add the S1 intersect S2 intersect S3 since I removed it one time: 7 + 5 = 12 and this is S2 - (S1 union S3). but then I'm lost. I can't implement algebra of sets for some reason, my mind refuses it.

dear daniton, thanks again for your answer. but my brain seems to crash with more than two sets. I can't understand the dynamics of three sets operations. I try to look at the diagram but I only seem to make it worse. for example, here is another problem that I completely couldn't understand. "in an exam there are 65 candidates, there are 3 sections. 5 candidates passed the three sections; all candidates that passed the third section have passed also the other two. 20 candidates passed only the first two sections, 3 candidates didn't pass any, and 50 passed the first section. how many candidates have passed only the first, and how many only the second?" so I extracted the data: Universe = 65 S1 = 50 S1 intersect S2 intersect S3 = 5 S1 intersect S2 = 20 complementary of S1 union S2 union S3 = 3 so I must find S1 - (S2 union S3) and S2 - S3 then I calculated S1 union S2 union S3 = 65 - 3 = 62 then S1 intersect S3 = 5 S1 union S3 = 55 and then I'm lost. I've made other calculations but I prefer not to post them here, since I don't understand them, even if the resulting numbers are correct.

hello thanks for the answer. the demonstration I have attempted of your example shows that there is an error: (A union B) intersect C seems not to be equal to (A intersect B) union (B intersect C). since A union B = {a; b; c; d; e; f; g} C = {d; f; h; i} so (A union B) intersect C = {d; f} but (A intersect B) union (B intersect C) = {d; e; f} since (A intersect B) = {d; e} and (B intersect C) = {d; f}

hello and thanks for your answer. I have tried to demonstrate what you suggested, but still I have some doubt about my reasoning. A union B union C = A+B+C - (A intersect B) - (B intersect C) - (A intersect C) + (A intersect B intersect C) I don't know what the + sign means in the set theory and I couldn't find any reference to that sign in set theory online, so I implemented it in a way that seems to be working for the example. consider A = {a; b; c; d; e} B = {d; e; f; g} C = {d; f; h; i} A intersect B = {d; e} A intersect C = {d} B intersect C = {d; f} A intersect B intersect C = {d} so A+B+C = {a; b; c; d; e; d; e; f; g; d; f; h; i} as you see I considered double entries for some elements, since the + operator probably acts differently from the union operator. (A+B+C) - (A intersect B) = {a; b; c; d; e; f; g; d; f; h; i} (A+B+C) - (A intersect B) - (B intersect C) = {a; b; c; d; e; f; g; h; i} (A+B+C) - (A intersect B) - (B intersect C) - (A intersect C) = {a; b; c; e; f; g; h; i} and finally (A+B+C) - (A intersect B) - (B intersect C) - (A intersect C) + (A intersect B intersect C) = {a; b; c; e; f; g; h; i; d} is this correct? thanks in advance

hello. I have a doubt about my thought process on this exercise. we have 3 sets: A,B,C. A intersection B intersection C = 10% A intersection C = 20% B intersection C = 20% A intersection B = 40% A = 70% B = 55% C = 30% what is the complement (A union B union C)? first I determined that: U = 100% then I cut the problem in order to calculate A union B first. A union B = (70+55)-40 = 85 (adding A and B minus common elements, to find the true content of A union B) (A union B) union C = (85+30)-20-20 = 75 (same procedure, removing the common A intersection C and B intersection C) then I add A intersection B intersection C to the last calculations: 75 + 10 (A intersection B intersection C) - and this is the point that I'm not sure about. I add the (A intersection B intersection C) because in that triple intersection lies part of C, that is not present in the intersection between (A union B) intersection C. but I'm not sure that this is the right way to do it... the result is correct: complement (A union B union C) = U - (A union B union C) = 15% but I'm not satisfied wity my reasoning at all. it sounds cheesy. what do you think about it? thanks in advance.

hello, I'm studying mathematics by myself, and following some books I came across this exercise: "in a class of 28 students, 12 practice swimming, 8 practice soccer, 7 practice both swimming and soccer. how many students do not practice any of these sports?" so then I extracted the data: Universe = 28 Swimming = 12 Soccer = 8 Sw intersection So = 7 Universe - (Sw union So) = complement (Sw union So) = ? then I calculated arithmetically: (12+8)-7 = 13 (Sw union So) 28-13=15 (complement (Sw union So) ) <- solution of the exercise I don't know if it is the right approach to solve the problem, even if the solution is correct. how can I solve this with set theory instead of arithmetics? thanks in advance

thanks so much =)

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.

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.