Exercises for self study. Complete these exercises in the textbook and
compare your solutions against those in the back of the book. These exercises
will not be graded.
p. 132, Exercise 19
p. 144, Exercises 15 and 17
p. 162, Exercise 21
Exercises to be graded. Complete these exercises and turn in your
solutions.
For sets $A$ and $B$, is $A \cap (A \cup B) = A$? If true, prove the
equality. If false, give a counterexample.
Give examples of sets $A$ and $B$ such that $A \in B$ and
$A \subseteq B$. Briefly justify your claim.
For the set of all integers ${\bf Z}$, for each of the following
properties, give an example of a function
$f : {\bf Z} \rightarrow {\bf Z}$ in a single formula that
satisfies the given property. Prove your claim.
(a) One-to-one but not onto.
(b) Onto but not one-to-one.
(c) One-to-one correspondence, different from the identity funcion
$f(n) = n$ for $n \in {\bf Z}$.
(d) Neither one-to-one nor onto.
(a) Prove that, given finite sets $A$ and $B$ of the same order, if
$f : A \rightarrow B$ is a one-to-one function, then $f$ is also onto.
(b) Give a counterexample, a one-to-one function, to demonstrate that
(a)'s conclusion is false when given sets are infinite sets. Prove your
claim.
Plagiarism and academic dishonesty.
Remember, under any circumstance, you must not copy part or all of another's
work and present it as your own. For more details, read
our course policy on plagiarism and
academic dishonesty.
