Page 43 - Calculus Demystified

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SOLUTION CHAPTER 1 Basics
We have
2
sin θ
2
tan θ + 1 = + 1
2
cos θ
2
2
sin θ cos θ
= +
2
2
cos θ cos θ
2
2
sin θ + cos θ
=
2
cos θ
1
=
2
cos θ
(where we have used Property (1))
2
= sec θ.
You Try It: Use identities (11) and (12) to calculate cos(π/12) and sin(π/12).
1.8 Sets andFunctions
We have seen sets and functions throughout this review chapter, but it is well to
bring out some of the ideas explicitly.
A set is a collection of objects. We denote a set with a capital roman letter, such
as S or T or U.If S is a set and s is an object in that set then we write s ∈ S and
we say that s is an element of S.If S and T are sets then the collection of elements
common to the two sets is called the intersection of S and T and is written S ∩ T .
The set of elements that are in S or in T or in both is called the union of S and T
and is written S ∪ T .
A function from a set S to a set T is a rule that assigns to each element of S a
unique element of T . We write f : S → T .
EXAMPLE 1.21

Let S be the set of all people who are alive at noon on October 10, 2004
and T the set of all real numbers. Let f be the rule that assigns to each
person hisor her weight in poundsat precisely noon on October 10, 2004.
Discuss whether f : S → T isa function.

SOLUTION
Indeed f is a function since it assigns to each element of S a unique element
of T . Notice that each person has just one weight at noon on October 10, 2004:
that is a part of the definition of “function.” However two different people may
have the same weight—that is allowed.