Tuesday, August 26, 2014

Section 2-6: Unions and Intersections of Figures (Day 14)

Section 2-6 of the U of Chicago text focuses on unions and intersections. This is, of course, the domain of set theory.

In many ways, set theory is the basis of modern mathematics, and so many textbooks -- including higher math such as Precalculus and beyond -- mention set theory early on. Of course, the focus in this text is on unions and intersections of geometric figures. In particular, unions are used to define both polygon and angle, while intersections are used to define parallel lines.

The first three examples in the text, where the underlying sets contain natural numbers, real numbers, and points -- are OK. But I didn't like the fourth example, on airlines. I've decided to throw this one out -- if we want a non-mathematical example, why not just use sets of letters, such as {a, e, i, o, u}, the set of vowels?

One of the most important sets in mathematics is the null set, or empty set. According to the text, this set can be written as either { } or an O with a line through it (often called O-slash by students). Once again, since I can't represent that symbol on Blogger, let's use the strikethrough instead: O.

Now the text mentions that the intersection of two sets might be the empty set. But it doesn't mention what happens when one finds the union, or intersection, of the empty set and another set. As it turns out, the union of the empty set and another set is that other set -- so the empty set acts as the identity element for union, just as 0 is the identity for addition and 1 is the multiplicative identity. But the intersection of the empty set and another set is the empty set -- so the empty set acts as the absorbing element for intersection, just as 0 is the absorbing element for multiplication.

One question students often ask is, if { } is the empty set and O is the empty set, what's {O}? When I was young, I once heard a teacher point out that this is not the empty set because it's no longer empty -- it contains an element.

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