By Bradley Jackson, Dmitri Thoro

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At a dinner party there are 8 guests. The dinner takes place at a table shaped like a regular octagon. Each edge has one place setting that is labeled with the name of a different guest. Originally each person sits in the wrong place. Explain why the table can be rotated so that at least two persons are sitting in the right place. 7. a) The 10 digits {0, 1,2,. , 9} are arranged in a straight line. No matter how they are arranged, show that there are always three consecutive digits . whose sum is at least 13.

If the two dice are different, say, one is colored green and the other is colored red, then how many different outcomes are possible when the two dice are rolled? Figure I . I. 1 I I I5 Solution Think of recording each outcome as an ordered pair, the green die in the first coordinate and the red die in the second coordinate. Since there are six outcomes for each die, the total number of outcomes equals the total number of ordered pairs where each digit is an element of {l,2,3,4,5,6}. Therefore, the total number of outcomes is 6 .

In terms of ordered pairs, the function f can be thought of as a set of ordered pairs in A x B, where each a in A is the first coordinate of exactly one ordered pair. If (a, b) is in f, we also write that f(a) = b and say that b is the image of a. A function can also be thought of as an assignment or distribution of the objects in A to the objects in B, where each object of A is assigned to exactly one object in B. In general, each b in B is the image of zero, one, or more elements of A. Types of Functions We frequently encounter several types of special functions.