lec0320 - Topics Covered on Exam Relations: definition and...

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Topics Covered on Exam Relations: definition and graphical view, binary and n-ary relation, inverse, composition of a relation, associativity of composition, reflexive, irreflexive, symmetric, anti-symmetric, and transitive properties, equivalence relations, partial ordering relations, reflexive, symmetric, and transitive closures Functions: definition and graphical view, composition of a function, inverse, injection, surjection, and bijection. Sections in the book that will be useful: Chapter 5: Sections 1, 2, 6 Chapter 7: Sections 1, 4 Also, since a relation is simply a set, it may be useful to review material on sets from chapter 3. How to study: First, flip through my notes, making sure you understand the examples presented. Look at the past couple homework solutions, and also look at previous homework questions on this material. Practice these problems. Finally, flip through the book to make sure you understand what is in each of the sections I mentioned above. Keep in mind that I may not have covered everything in these sections.
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Format Unlike last time I will actually have some T/F and multiple choice/matching, as well as some short answer. However, as always, I will have a few proofs of the nature that you have seen over the past three or four weeks. The key to these proofs is to first understand what the question is asking. Once you have determined that, apply the pertinent definitions in such a way that you can show the assumptions imply the conclusion. Some specific things to remember: 1) A relation can be defined in two ways: either explicitly listed such as R = {(1,2), (2,3)} or generally as the following: R = {(a,b) | a Z + b Z + b = a + 1} Make sure you understand how to interpret both. Example: R = {(a,b) | a Z + b Z + 5 c | a = b + 2c, where c Z} This is read as: “The set R contains all elements (a,b) such that a and b are positive integers and there exists an integer s such that a = b + 2c.” In essence, the value of c has little to do with the membership in the set.
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2) Know how to use the definition of relation/function composition. Consider the situation where R, S, and T are all relations over the set A x A. For example, if we know that (x,y) (R ° S) ° T, we can deduce that (x,z) R ° S and (z, y) T, for some element z in A. Similarly, if (x,z) R ° S, then we know that (x,w) R and (w,z) S, for some element w of A. Now why is it that I used a z one time in applying the defintion and a w the next time? The
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lec0320 - Topics Covered on Exam Relations: definition and...

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