Understanding the Transitive, Reflexive, and Symmetric PropertiesDate: 06/30/2008 at 21:20:22 From: Heather Subject: Equivalence Relations I am trying to help a person studying to be an elementary school teacher. I am helping her with a math class, however I have gotten stuck on the transitive, reflexive and symmetrical properties. For example: We need to classify each of the following math relationships on the specific sets as reflexive, symmetrical, or transitive. Determine whether each is an equivalence. a) Is not equal to (counting numbers) b) is less than (counting numbers) c) has the same area as (triangles) I understand what each property means, and I think I understand it when there is an actual set of numbers (such as {(1,1), (2, 3) etc}. I have run into some questions that seem more vague to me and am having a hard time applying the properties. I guess I'm most confused because I'm not sure I understand what they mean with the given set. Going with letter a: When I see this, I think that for not equal to it would not be reflexive since it says not equal to, it might be symmetrical because if a is not equal to counting numbers then counting numbers are not equal to a, and it would not be transitive because to be transitive a is equivalent to b and b is equivalent to c so a is equivalent to c and I don't see how I could get 3 sets of numbers from this problem. Date: 06/30/2008 at 23:14:18 From: Doctor Peterson Subject: Re: Equivalence Relations Hi, Heather. It looks like you are being confused by the fact that the properties are often stated too briefly, not making clear what the x, y, and z (or whatever letters are used) refer to. Here's a statement of the definitions: Relation R on a set S is reflexive if for any x in S, x R x. Relation R on a set S is symmetric if for any x and y in S such that x R y, it is also true that y R x. Relation R on a set S is transitive if for any x, y, and z in S such that x R y and y R z, it is also true that x R z. Note that my x, y, and z are elements of the set S on which the relation is defined. They are not entire sets, or anything like that. Also, what is said must be true for ANY x, y, and z. (By the way, what I'm writing as x R y should be read not as "x is equivalent to y" but as "x is related to y by R"; we don't know yet that R is an equivalence relation, only that it is a relation. You may be taught a different notation; let me know and I can rewrite this so it looks more familiar.) Let's look at example (a): Is not equal to (counting numbers) I'm assuming that "counting numbers" means the natural numbers 1, 2, 3, ... , though that doesn't make much difference here. First, is this relation reflexive over the counting numbers? By the definition, we are asking whether, for ANY counting number x, it is true that x is not equal to x Clearly, this is NEVER true, so the relation is not reflexive. Some relations fail to be reflexive only because SOME numbers are not "related" to themselves, though others are. Is it symmetric? Here we are asking whether, for ANY TWO counting numbers x and y for which x is not equal to y it is ALWAYS true that y is not equal to x Well, we know very well that those two statements mean the same thing; the fact that, say, 3 is not equal to 5 means the same as that 5 is not equal to 3. So it IS symmetric. How about the transitive property? That says that for ANY THREE counting numbers x, y, and z for which x is not equal to y, and y is not equal to z then it MUST be true that x is not equal to z For example, it is true that 3 is not equal to 5, and 5 is not equal to 7 and it is also true that 3 is not equal to 7 So that example fits; and many others will too. But is this ALWAYS true? Well, nothing in the definition says that x, y, and z all have to be different elements; how about this case? 3 is not equal to 5, and 5 is not equal to 3 Is it true that 3 is not equal to 3? NO! It can be hard sometimes to find a "counterexample" like this; you have to think carefully about what it might take to make the statement false. In this case, I asked myself for an example of x and z that are NOT "not equal", so the conclusion of the definition would be false, and then chose a y that would make the condition true. So we've found that this relation is symmetric, but not reflexive or transitive. It is not an equivalence relation. That shouldn't be too surprising; we wouldn't expect "not equals" to behave the same as "equals" (which is what "equivalence relation" means); it's more opposite to "equals". What's interesting is to see in what ways it IS the same, and in what ways it is not. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 07/03/2008 at 18:03:55 From: Heather Subject: Thank you (Equivalence Relations) Thank you so much! I wish the book had put it that way, the other problems made so much more sense after your explanation. Thanks again! |
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