Algebra Know-It-ALL

Once in a while, you’ll see a three-barred equals sign to indicate that two sets are congruent. In this case, we would write E≡F ...

Overlapping sets When two sets have at least one element in common, they are called overlapping sets. In formal texts you might ...

U as the set of all the points on or inside the oval. When you have two overlapping sets, one of them can be a subset of the oth ...

Intersection of two nonempty disjoint sets When two nonempty sets are disjoint, they have no elements in common, so it’s impossi ...

Here, the intersection set has four elements: W 3 −∩W0+= {0, 1, 2, 3} Figure 2-4 is a Venn diagram that shows two overlapping se ...

Solution There are countless examples of set pairs like this. Let’s look at the set of all positive whole numbers divis- ible by ...

And for the null set ∅ ∪ ∅ = ∅ When two sets are congruent, their union is the same as their intersection. This might seem trivi ...

32 The Language of Sets Universe XUY X Y Figure 2-5 Two overlapping sets, X and Y. Their union is shown by the entire shaded reg ...

outside of the rectangle and part of it is the outside of the oval. Any element inside the region bounded by the dashed line is ...

What is the intersection of these two sets? What is their union? A= {1, 1/2, 1/3, 1/4, 1/5, 1/6, ...} G= {1, 1/2, 1/4, 1/8, 1/ ...

The whole numbers starting with 0 and counting upward are usually called the natural num- bers. Some mathematicians don’t call 0 ...

Building new numbers Let’s make up a rule that we can use to generate new natural numbers, one after another. Suppose we’ve buil ...

A definition for infinity When we write out any particular natural number p in the form of a set of smaller numbers, that set ha ...

about them, there are plenty of Web sites you can explore. Go to your favorite search engine and enter “infinite ordinal” or “tr ...

Even numbers Aneven natural number is a whole number whose numeral ends in 0, 2, 4, 6, or 8. If you multiply every number in the ...

40 Natural Numbers and Integers We can’t break the number 3 down into a product of natural numbers other than itself and 1. How ...

the quotient are both factors of the original number. Sometimes the quotient will be prime, and sometimes it won’t be. If it isn ...

42 Natural Numbers and Integers The best way to find out whether or not a large odd number is prime is to try to factor it into ...

Divisibility If you want to know whether or not a large number can be divided by a single-digit number without leaving a remaind ...

44 Natural Numbers and Integers display—but it will be finite. Let’s call it y. What if we add 1 to y, getting a number even lar ...