By Sheldon Axler
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Extra resources for Algebra and Trigonometry
Example 2 Write (1, 5) ∪ (3, 7] as an interval. solution The figure here shows that every number in the interval (1, 7] is either in (1, 5) or is in (3, 7] or is in both (1, 5) and (3, 7]. The figure shows that (1, 5) ∪ (3, 7] = (1, 7]. Similarly, the union of three or more sets is the collection of objects that are contained in at least one of the sets. 1 5 3 7 30 chapter 1 The Real Numbers The next example goes in the other direction, starting with a set and then writing it as a union of intervals.
Xb Finally, we conclude this subsection by recording some identities involving fractions and additive inverses: When faced with complicated expressions involving fractions that are themselves fractions, remember that division by a fraction is the same as multiplication by the fraction flipped over. 24 chapter 1 The Real Numbers Fractions and additive inverses −a a a = =− b −b b −a a = −b b a c ad − bc − = b d bd Symbolic Calculators The symbol ≈ means “approximately equal”. example 9 Over the last several decades, inexpensive electronic calculators and computers drastically changed the ease of doing calculations.
Similarly, the union of three or more sets is the collection of objects that are contained in at least one of the sets. 1 5 3 7 30 chapter 1 The Real Numbers The next example goes in the other direction, starting with a set and then writing it as a union of intervals. example 3 Write the set of nonzero real numbers as the union of two intervals. solution The set of nonzero real numbers is the union of the set of negative numbers and the set of positive numbers. In other words, the set of nonzero real numbers equals (−∞, 0) ∪ (0, ∞).