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little more careful to maintain our data structure in good shape As any housekeeper knows, a little extra effort put into routine maintenance can pay off handsomely in the long run, by forestalling major calamities We have in mind a particular maintenance operation for our union- nd data structure, intended to keep the trees short during each find, when a series of parent pointers is followed up to the root of a tree, we will change all these pointers so that they point directly to the root (Figure 57) This path compression heuristic only slightly increases the time needed for a find and is easy to code function find(x) if x = (x) : (x) = find( (x)) return (x) The bene t of this simple alteration is long-term rather than instantaneous and thus necessitates a particular kind of analysis: we need to look at sequences of find and union operations, starting from an empty data structure, and determine the average time per operation This amortized cost turns out to be just barely more than O(1), down from the earlier O(log n) Think of the data structure as having a top level consisting of the root nodes, and below it, the insides of the trees There is a division of labor: find operations (with or without path 137

your research on stocks with the best ratings, she says Our researchers have found that stocks that outperform during a poor market are frequently the ones that make big price gains when the market s rally resumes In a downturn, Stalter suggests creating a watch list of fundamentally and technically strong stocks that could outperform when the market turns around When the market does turn around, be very choosy about which stocks to buy, and be ready to cut them quickly if market action is sputtering, she cautions Once again, Stalter says, the keys to determining if you re in an uptrend or downtrend are price and volume Professional investors account for the bulk of market action, she points out Institutions drive market direction Stalter has some final advice: It takes a bit of time to master those rules, but it s worth it And it pays to control emotions like fear and greed, so that s where having a set of buy and sell rules helps That way, you can operate with a logical plan you adhere to in every market scenario It s a whole lot better than relying on your own emotions, or even worse, listening to tips and opinions that you hear on TV

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Question 3-1

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Answer 3-1

We can define the number 0 as the set containing nothing That s the null set: 0={}= Once we ve defined the number 0, then we can define the number 1 as the set containing the number 0, like this: 1 = {{ }} = { } = {0} After that, we can build the rest of the natural numbers upon each other: 2 = {0, 1} 3 = {0, 1, 2} 4 = {0, 1, 2, 3} n + 1 = {0, 1, 2, , n} and so on, forever

Kate Stalter is a markets writer for Investor s Business Daily and host of the Market Wrap video at Investorscom Since 2001, she has edited and written for IBD columns The Real Most Active, Stocks in the News, and Investor s Corner Stalter has cohosted Webcasts with TDAmeritrade and regularly presents at IBD investing seminars nationwide She is an in-demand commentator on national radio programs She received her MBA from the Kellogg School of Management at Northwestern University

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