Intersection, Difference, UnionDate: 12/18/2002 at 21:10:00 From: Phoebe Subject: Algebra 1 I do not understand 'members of both A but not of B and members of both A and B'. My math book does not introduce these sorts of problems. All it covers is sets and subsets, rational numbers, whole mumbers, etc. Example: A = {-3, -2, -1/2, 0, 1, 3} B = {-2, 1/2, 1, 3/2, 2, pi} The problem: integers that are members of A but not of B, integers that are members of both A and B, integers that are members of either A or B. Integers of A but not of B: does this mean set A only and not B? Date: 12/20/2002 at 11:54:03 From: Doctor Ian Subject: Re: Algebra 1 Hi Phoebe, Suppose we line up the sets like this: A = { -3, -2, -1/2, 0, 1, 3 } B = { -2, 1/2, 1, 3/2, 2, pi } To find the elements that are in both A and B, we can see which ones in A have a matching element from B: * * A = { -3, -2, -1/2, 0, 1, 3 } B = { -2, 1/2, 1, 3/2, 2, pi } So elements in both A and B = { -2, 1 } Note that we get the same thing if we look at elements of B, and see which ones have a matching element in A: A = { -3, -2, -1/2, 0, 1, 3 } B = { -2, 1/2, 1, 3/2, 2, pi } * * So elements in both A and B = elements in both B and A This is called the 'intersection' of the sets. To find elements that are in A, but not in B, we look for elements where there is no match: * * * * A = { -3, -2, -1/2, 0, 1, 3 } B = { -2, 1/2, 1, 3/2, 2, pi } So elements in A but not in B = { -3, -1/2, 0, 3 } In this case, the order _does_ matter. If we look for elements that are in B, but not in A, we get a different set: A = { -3, -2, -1/2, 0, 1, 3 } B = { -2, 1/2, 1, 3/2, 2, pi } * * * * So elements in B but not in A = { 1/2, 3/2, 2, pi } We call the set of elements that are in A but not B the 'difference' of A and B. You can think of it this way: To find the difference A-B, you start with the elements of A, and remove any elements that also happen to be in B. To find the elements that are in A or B, we just take all the elements in both sets: A = { -3, -2, -1/2, 0, 1, 3 } B = { -2, 1/2, 1, 3/2, 2, pi } * * * * * * * * * * So elements in A or B = { -3, -2, -1/2, 0, 1/2, 1, 3/2, 2, 3, pi } This is called the 'union' of the sets. Now, in your case, you were asked to find the _integers_ that survived these operations. So you'd want to rule out anything that ended up in any of your result sets that isn't an integer, e.g., elements in A or B = { -3, -2, -1/2, 0, 1/2, 1, 3/2, 2, 3, pi } integers in A or B = { -3, -2, 0, 1, 2, 3 } In effect, you're finding an intersection, difference, or union of A and B; and then you're finding the intersection of that with the set of all integers. Does that make sense? When you're first getting used to this, it's easier to think in terms of sets that are more meaningful than collections of numbers or letters. For example, Actors = { Julia Roberts, Tom Hanks, Helena Bonham Carter, Brad Pitt } Women = { Julia Roberts, Helena Bonham Carter, Sally Ride, Valerie Shute } Now let's think about those same operations: Operation What it means Result ---------------- ---------------- ----------- Intersection Actors who are Julia Roberts also women Helena Bonham Carter Difference Actors who are Tom Hanks (actors - women) not women Brad Pitt Difference Women who are Sally Ride (women - actors) not actors Valerie Shute Union People who are Julia Roberts actors or women Tom Hanks Helena Bonham Carter Brad Pitt Sally Ride Valerie Shute Note that the use of the word 'or' is a little tricky. In everyday English, we usually use 'or' to mean that something is in one of two mutually exclusive conditions. For example, we say that it's raining OR it's not. We say that a person is male OR female. And so on. But that's not how we use it in set theory. In set theory, when we say that something is in set A or set B, we mean that it's in A, or in B, or in both sets. If we want to specify the elements that are in either set but not the other, we have to be explicit about that: "in either A or B, but not both." To simply say that something is "in either A or B" without adding "but not both" is somewhat ambiguous, because it's not clear which meaning is intended. How would we find the people who are either actors, or women, but not both? One way would be to subtract the women from the actors, (actors - women) = { Tom Hanks, Brad Pitt } and then subtract the actors from the women, (women - actors) = { Sally Ride, Valerie Shute } and then take the union of these: (women - actors) U (actors - women) This is called a 'symmetric difference'. So we can add that to our table: Operation What it means Result ---------------- ---------------- ----------- Intersection Actors who are Julia Roberts also women Helena Bonham Carter Difference Actors who are Tom Hanks (actors - women) not women Brad Pitt Difference Women who are Sally Ride (women - actors) not actors Valerie Shute Union People who are Julia Roberts actors or women Tom Hanks Helena Bonham Carter Brad Pitt Sally Ride Valerie Shute Symmetric People who are Tom Hanks Difference either actors, Brad Pitt or women, but Sally Ride not both Valerie Shute Anyway, the whole business with 'or' takes some getting used to, but these are the main ideas. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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