DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

You’ll prove the Sum Principle for any finite number of blocks in the
next chapter. For right now you may accept it as true and use it wherever
you like. Of course you must be careful that your proofs in the next chapter
do not depend on any results which you’ve proved using the general Sum
Principle.
◦15. In a biology lab study of the effects of basic fertilizer ingredients on
plants, 16 plants are treated with potash, 16 plants are treated with
phosphate, and a total of eight plants among these are treated with
both phosphate and potash. No other treatments are used. How many
plants receive at least one treatment? If 33 plants are studied, how
many receive no treatment?
◦16. Use partitions to prove a formula for the size |A∪B| of the union A∪B
of any two (finite but not necessarily disjoint) sets A and B in terms
of the sizes |A| of A, |B| of B, and |A ∩ B| of the intersection A ∩ B.
The formula you proved in the last problem is a special case of the Principle
of Inclusion and Exclusion, which is considered more thoroughly
in Chapter 6.
1.3 Functions and their Directed Graphs
A typical way to define a function f from a set S (called the domain of the
function) to a set T (which in discrete mathematics is commonly referred to
as its co-domain) is: A function f is a relation from S to T which relates
each element of S to one and only one member of T . The notation f(x)
is used to represent the element of T that is related to the element x of S,
and the standard shorthand f : S → T is used for “f is a function from
S to T ”. Please note that the word “relation” has a precise meaning in
mathematics. Do you know it? Refer to Appendix A if you need a review
of this terminology.
Relations between subsets of the set of real numbers can be graphed in
the Cartesian plane, and it is helpful to remember how you used a graph of
a relation defined on the real numbers to determine whether it is actually a
function. Namely, to do this you learned that you should check that each
vertical straight line crosses the graph of the relation in at most one point.
You might also recall how to determine whether such a function is one-to-one
by examining its graph. If each horizontal line crosses the graph in at most
one point, the function is one-to-one. If even one horizontal line crosses the
graph in more than one point, the function is not one-to-one.
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