Let S = {1,2,3}, and use the ordered-pair definition of a relation to give examp

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Let S = {1,2,3}, and use the ordered-pair definition of a relation to give examples of the 8 types of relations on S specified by the properties ( := Reflexive, :=Symmetric, and :=Transitive) listed in the following table (where Y := Yes and N := No): Then give an example from mathematics or from everyday life illustrating each of the 8 cases listed above. For example [and please understand that these particular examples are now off limits], for (1) one might let S be the set of all human beings and for x,y S define xRy x has the same blood type as y. Or, one might let S = P(X) = the set of all subsets of some finite set X , and for z,y S define xRy |x| = |y|, i.e., the number of elements in x equals the number of elements in y. In giving an example for Case (5), above, you will have to clarify for yourself (and me) how it is even possible to have this situation arise; in other words, you will have to explain what is wrong with the following argument which purports to prove that a symmetric, transitive relation must automatically be reflexive: but aRb and bRa rightarrow aRa, and so aRa, Recall that a relation R on a set S is a subset R of S times S. If a,b S, and (a,b) R, then we say that a is R-related to b or that a is related to b with respect to the relation R or that a is related to b under the relation R, and we write aRb (a,b) R. Then, R is reflexive R is symmetric if aRb, then bRa; finally, R is transitive if aRb and bRc, then aRc.

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