Essential Discrete Mathematics by Todd Feil

By Todd Feil

This ebook introduces readers to the math of desktop technological know-how and prepares them for the mathematics they're going to come upon in different university classes. It comprises functions which are particular to desktop technological know-how, is helping beginners to improve reasoning abilities, and offers the elemental arithmetic helpful for desktop scientists. bankruptcy themes contain units, capabilities and kin, Boolean algebra, normal numbers and induction, quantity concept, recursion, fixing recurrences, counting, matrices, and graphs. For desktop scientists and the enhancement of programming abilities.

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This is easy to show. A partition of a set is a collection of disjoint subsets whose union is the entire set. The equivalence class for = that contains a (let's call this set Ea) is all the b C A such that a = b. First note that every element a of A is in some class, namely Ea, since -= is reflexive. Now all we must show is that if EL and Eb overlap, they are identical. But ifc e EanEb, then a- cand b- c. But then c- b (since = is symmetric) and so a -=b (since - is transitive). That is, b E Ea.

F. find a relation on B that is a equivalence relations. Find one that is a partial order. Find one that is a total order. 19. Can an equivalence relation be an order relation? Why or why not? 20. Can an order relation be an equivalence relation? Why or why not? 21. Is every function a relation? Why or why not? 22. Is every relation a function? Why or why not? 23. Give an example of a relation that is reflexive, but not symmetric and not transitive. 24. Give an example of a relation that is symmetric, but not reflexive and not transitive.

But b E EL and so a = b. ) A similar argument shows that Ea C Eb and so Ea = Eb. The simplest equivalence relation "equals" on any set has rather trivial equivalence classes. Indeed, Ea = {a}, for element a. At the other extreme, consider the equivalence relation on any set where every element is related to every other element. How many equivalence classes are there for this relation? Chap. 2 Functions and Relations 35 Another special type of relation is one that is reflexive, antisymmetric, and transitive.

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