2.5: Sets are not ordered pairs (or tuples)

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You’ll remember from high school algebra the notion of an ordered pair \((x,y)\). We dealt with those when we wanted to specify a point to plot on a graph: the first coordinate gave the distance from the origin on the x-axis, and the second coordinate on the y-axis. Clearly an ordered pair is not a set, because as the name implies it is ordered: \((3,-4) \neq (-4,3)\). For this reason, we’ll be very careful to use curly braces to denote sets, and parentheses to denote ordered pairs.

By the way, although the word “coordinate" is often used to describe the elements of an ordered pair, that’s really a geometry-centric word that implies a visual plot of some kind. Normally we won’t be plotting elements like that, but we will still have use to deal with ordered pairs. I’ll just call the constituent parts “elements" to make it more general.

Three-dimensional points need ordered triples \((x,y,z)\), and it doesn’t take a rocket scientist to deduce that we could extend this to any number of elements. The question is what to call them, and you do sort of sound like a rocket scientist (or other generic nerd) when you say tuple. (Some people rhyme this word with “Drupal," and others with “couple," by the way, and there seems to be no consensus). If you have an ordered-pair-type thing with 5 elements, therefore, it’s a 5-tuple (or a quintuple). If it has 117 elements, it’s a 117-tuple, and there’s really nothing else to call it. The general term (if we don’t know or want to specify how many elements) is n-tuple. In any case, it’s an ordered sequence of elements that may contain duplicates, so it’s very different than a set.