lec0301 - Examples of Identifying Properties in Relations...

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Examples of Identifying Properties in Relations Is the following relation reflexive, irreflexive, symmetric, antisymmetric, or transitive? R = {(a,b) | a,b Z + a, 2a, and b are side lengths of a triangle} Note: For all triangles, the sum of the lengths of any two sides must exceed the length of the third side. Reflexive? No – because (a,a) R, this is because a triangle can not have sid lengths, a, a and 2a. Irreflexive? Yes – the previous argument holds for all positive integers a. Symmetric? No – (a, 2a) R, since we can have a triangle with side lengths a, 2a and 2a. However, (2a, a) R because we can not have a triangle with side lengths 2a, 4a and a. Antisymmetric? Yes – If we have a b, then we have (a,b) R. To prove this, consider a forming a triangle with side lengths a, 2a, and b. We know that we must have a+b > 2a for a triangle to be formed. BUT, a+b a+a = 2a, which means that a+b is NOT greater than 2a. Thus, in this situation, we have (a,b) R. Thus, for any element (a,b) R, we must have a < b. For each of these elements, we can guarantee that (b,a) R since b > a. Transitive? No – (a, 2a) R as shown above, and we also know that (2a, 4a) R, by a similar analysis. But, we can show that (a,4a) R because a triangle can not have side lengths a, 2a and 4a.
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Consider the following relation: R = { (a,b) | a Z + b Z + ab = c 2 for some positive integer c} Prove that it is an equivalence relation. Reflexive? Yes – (a,a) R because a 2 = c 2 , when c is equal to a, a positive integer. Symmetric? Yes – if (a,b)
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lec0301 - Examples of Identifying Properties in Relations...

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