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A mathematical statement of the form" A if and only if B" is true if whenever A is true then so is B and whenever A is false then so is B. " 1:11 A LOOK BACK We began this chapter with a particular problem (admittedly one of only small import) and introduced some substantial mathematics in order to understand the problem and its solutions. There are a number of ideas that you should be comfortable with before you proceed to the next chapter. Foremost among the important concepts discussed is the notion of algorithm.

C) J3(X, y,:) = ~ (x v Y v z). (d) 14(X, Y,:) = ~ (x A Y A:). '(' y, z) = (x A y) V (x A z). (f) 16(X, y, z, W) = (x A y) v (z A w). 8. A function is said to be in disjunctive normal form (or DNF) if it is in the form where, for i = 1,2, ... ,k, Di is of the form L[ A L2 A ... A L j with each Li a literal for i = 1,2, ... , j. Find all functions in Exercises 3 and 7 that are in DNF. 9. Find an example of a function that is in neither CNF nor DNF. 10. For each of the following, find an equal function expressed in DNF.

Typically, we describe the set by where a 1,Ll2"" ,an stand for the 11 elements in A. These may be the numbers ~ 1, 2, ... ,ilL or they may stand for a different set of numbers like : 1,5,17,25, ... ,94)' or they may stand for names of pieces of fruit as in : apple, banana.... , quince]. Here are two methods to list all subsets of a general set like A. Problem. Given an Il-set A = (Ll l' Ll2,' .. , an;' list all 2n subsets of A. Response 1 STEP 1. List the empty set STEP 2. Set j : = 1 STEP 3. List all subsets of A that contain j elements STEP 4.

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A note on an exponential semilinear equation of the fourth order by Mugnai D.

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