Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
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134 A. Relations<br />
Problem 329. Here are some questions that will help you get used to the<br />
formal idea of a relation and the related formal idea of a function. S will<br />
stand for a set of size s and T will stand for a set of size t.<br />
(a) What is the size of the largest relation from S to T?<br />
(b) What is the size of the smallest relation from S to T?<br />
(c) The relation of a function f : S → T is the set of all ordered pairs<br />
(x, f (x)) with x ∈ S. What is the size of the relation of a function<br />
from S to T? That is, how many ordered pairs are in the relation of a<br />
function from S to T? (h)<br />
(d) We say f is a one-to-one function or injection from S to T if each<br />
member of S is related to a different element of T. How many different<br />
elements must appear as second elements of the ordered pairs in the<br />
relation of a one-to-one function (injection) from S to T?<br />
(e) A function f : S → T is called an onto function or surjection if each<br />
element of T is f (x) for some x ∈ S What is the minimum size that S<br />
can have if there is a surjection from S to T?<br />
Problem 330. When f is a function from S to T, the sets S and T play a big<br />
role in determining whether a function is one-to-one or onto (as defined in<br />
Problem 329). For the remainder of this problem, let S and T stand for the<br />
set of nonnegative real numbers.<br />
(a) If f : S → T is given by f (x) =x 2 ,is f one-to-one? Is f onto?<br />
(b) Now assume S ′ is the set of all real numbers and g : S ′ → T is given<br />
by g(x) =x 2 .Isg one-to-one? Is g onto?<br />
(c) Assume that T ′ is the set of all real numbers and h : S → T ′ is given<br />
by h(x) =x 2 .Ish one-to-one? Is h onto?<br />
(d) And if the function j : S ′ → T ′ is given by j(x) =x 2 ,isj one-to-one?<br />
Is j onto?<br />
Problem 331. If f : S → T is a function, we say that f maps x to y as another<br />
waytosaythat f (x) =y. Suppose S = T = {1, 2, 3}. Give a function from<br />
S to T that is not onto. Notice that two different members of S have mapped<br />
to the same element of T. Thus when we say that f associates one and only<br />
one element of T to each element of S, it is quite possible that the one and<br />
only one element f (1) that f maps 1 to is exactly the same as the one and<br />
only one element f (2) that f maps 2 to.
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