Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

Draw a Venn diagram for the symmetric difference of the sets A and B.

Find $\mathit{f}\mathbf{\left(}\mathit{n}\mathbf{\right)}$when $\mathit{n}\mathbf{=}{\mathbf{4}}^{\mathbf{k}}$, where $\mathit{f}$satisfies the recurrence relation$\mathit{f}\mathbf{\left(}\mathit{n}\mathbf{\right)}\mathbf{=}\mathbf{5}\mathit{f}\mathbf{(}\mathbf{/}\mathbf{4}\mathbf{)}\mathbf{+}\mathbf{6}\mathit{n}$, with$\mathit{f}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$.

If f and$\mathrm{f}\circ \mathrm{g}$ are one-to-one, does it follow that g is one-to-one? Justify your answer.

Compute each of these double sums

$\begin{array}{r}\sum _{\mathrm{i}=1}^3 \sum _{\mathrm{j}=1}^2 (\mathrm{i}-\mathrm{j})\\ \sum _{\mathrm{i}=0}^3 \sum _{\mathrm{j}=0}^2 (3\mathrm{i}+2\mathrm{j})\\ \sum _{\mathrm{i}=1}^3 \sum _{\mathrm{j}=0}^2 \mathrm{j}\\ \sum _{\mathrm{i}=0}^2 \sum _{\mathrm{j}=0}^3 {\mathrm{i}}^2{\mathrm{j}}^3\end{array}$

_{Show that} (0,1)_{and R have the same cardinality. [Hint: Use the Schroder-Bernstein theorem].}

Find\({{\bf{A}}^{\bf{3}}}\)if

(a) \({\bf{A = }}\left\{ {\bf{a}} \right\}\)

(b) \({\bf{A = }}\left\{ {{\bf{0,a}}} \right\}\)

Does the check digit of an ISSN detect every single error in an ISSN? Justify your answer with either a proof or a counterexample.

Show that the set of all finite subsets of the set of positive integers is a countable set.

Exercises 34–37 deal with these relations on the set of real numbers:

\({R_1} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a > b} \right\},\)the “greater than” relation,

\({R_2} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ge b} \right\},\)the “greater than or equal to” relation,

\({R_3} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a < b} \right\},\)the “less than” relation,

\({R_4} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \le b} \right\},\)the “less than or equal to” relation,

\({R_5} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a = b} \right\},\)the “equal to” relation,

\({R_6} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ne b} \right\},\)the “unequal to” relation.

35. Find

(a) \({R_2} \cup {R_4}\).

(b) \({R_3} \cup {R_6}\).

(c) \({R_3} \cap {R_6}\).

(d) \({R_4} \cap {R_6}\).

(e) \({R_3} - {R_6}\).

(f) \({R_6} - {R_3}\).

(g) \({R_2} \oplus {R_6}\).

(h) \({R_3} \oplus {R_5}\).

Question: If f and$\mathit{f}\mathbf{\circ }\mathit{g}$ are onto, does it follow that g is onto? Justify your answer.