• XeLaTeX (Unicode) or LaTeX (European Languages)
• Markup Language built on TeX
• Tags: \tag
• Environments: \begin{..} ... \end{..}
• Grouping: {...}
• Math mode: $$d = \sqrt{a_0^2 + a_1^2}$$
• Spacing: \vfil \hfil \bigskip \smallskip

• Font size: \tiny \small \normalsize \large \huge
• Font families: \bf \tt \sl \it
• Document segments: \section \subsection \subsubsection
• Graphics: \includegraphics{width=\columnwidth}{pic.png}
• Tables: \begin{tabular}{lr} aa & 3.5\\ b & 17.2\\ \end{tabular}
• Footnotes: \footnote{This is a note}
• Citations: \cite{Knuth:1970}
• Cross links: \label{id} .. As seen in \ref{id}

\addresslabel[\fboxsep=5mm]{\vbox to 75mm{%
    \hbox to 40mm{\kern-5pt
      $\vcenter{\hbox to 1.7cm{%
        \includegraphics[width=1.7cm]{payap.png}\hss}}\
        \vcenter{\raggedright
          Payap University\\ Faculty of Sci\\
          Amphur Muang\\  Chiang Mai 50000\\ 
          Thailand\\}$\hss}%
    \vfill{\centering {\Large\scshape 
       \textbf{Dr.~Robert~P.~Batzinger}\\}\medskip
       \textit{Instructor Emeritus}\\}
    \vfill{\raggedleft \small
        \textit{Office: \phonei}\\%
        \textit{LinkedIn: robert-batzinger}\\%
        \textit{Email: \emaili}\\}}}

• The order of the elements is meaningless.
  Given \( A = \{1,2,3\} \) and \( B = \{2,3,1\}: \) \( A = B \)
• There are no duplicate elements in a set.
• Given \( A = {1,2,3}: \) \( {\cal P}(A) = \) \( \{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\} \)

Please note which of these is true

• Given \( A=\{1,2,3\}, X = {\cal P}(A): \)
  \( \qquad\exists y \in X: y \subseteq A, y = A \)
  Hint (What is the value of \( y \) ?)

• Given \( A = \emptyset, X = {\cal P}(A):X = A \)
  Hint: (What is the value of \( X \) ?)

• Given $A = \emptyset, X = \

\[ \small\begin{array}{|c|c|c|c|c|} \hline A & |A| & {\cal P}(A) & count({\cal P}(A)) & |{\cal P}(A)| \\ \hline \emptyset & 0 & \{\emptyset\} & (1) & 1 \\ \hline \{1\} & 1 & \{\{\emptyset\},\{1\}\} & (1+1) & 2 \\ \hline \{1,2\} & 2 & \{\{\emptyset\}, \{1\}, \{2\}, \{1,2\}\} & (1+2+1) & 4 \\ \hline \{1,2,3\} & 3 & \{\{\emptyset\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\} & (1+3+3+1) & 8 \\ \hline \{1,2,3,4\} & 4 & \{\{\emptyset\}, \{1\}, \{2\}, \{3\}, \{4\}, \{1,2\}, \{1,3\}, \{1,4\}, & (1+4+6+4+1) & 16 \\ & & \{2,3\}, \{2,4\}, \{3,4\}, \{1,2,3\}, \{1,2,4\}, &&\\ & & \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\}\} & & \\ \hline \{1,2,3,4,5\} & 5 & \{\{\emptyset\}, \{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{1,2\}, \{1,3\}, \{1,4\}, \{1,5\}, &(1+5+10+10+5+1) & 32 \\ & & \{2,3\}, \{2,4\}, \{2,5\}, \{3,4\}, \{3,5\}, \{4,5\}, \{1,2,3\}, & & \\ & &\{1,2,4\}, \{1,2,5\}, \{1,3,4\}, \{1,3,5\}, \{1,4,5\}, \{2,3,4\}, & & \\ & & \{2,3,5\}, \{2,4,5\}, \{3,4,5\}, \{1,2,3,4\}, \{1,3,4,5\}, & & \\ & &\{1,2,4,5\}, \{1,2,3,5\}, \{2,3,4,5\}, \{1,2,3,4,5\}\}& & \\ \hline \end{array} \]

puts <<endmsg
A:#{setA.inspect} B:#{setB.inspect} 
C:#{setC.inspect} 
B subset A:    #{setB.subset?(setA)}
A superset B:  #{setA.superset?(setB)}
A union B:     #{setA.union(setC).inspect}
A intersect C: #{setA.intersection(setC).inspect}
A intersect C: #{(setA & setC).inspect}
A intersect C?:#{setA.intersect?(setC)}
A subtract C:  #{setA.difference(setC)}
Complement(B): #{setU.subtract(setB).inspect}
Cardinality(A):#{setA.size}
4 member of C?:#{setC.include?(4)
endmsg

A:#<Set: {1, 2, 3}>  B:#<Set: {2, 3}> 
C:#<Set: {2, 3, 4}> 
B subset A:     true
A superset B:   true
A union B:      #<Set: {1, 2, 3, 4}>
A intersect C:  #<Set: {2, 3}>
A intersect C:  #<Set: {2, 3}>
A intersect C?: true
A subtract C:   #<Set: {1}>
Complement(B):  #<Set: {1, 4}>
Cardinality(A): 3
4 member of C?: true

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{venndiagram}
\usepackage{xcolor}
\begin{document}\Large
\begin{venndiagram3sets}[labelOnlyA={1},
labelOnlyB={2},labelOnlyC={3},
labelOnlyAB={4},labelOnlyAC={5},
labelOnlyBC={6},labelABC={7},
labelNotABC={8},shade=orange]
\fillACapB \fillBCapC \fillACapC
\end{venndiagram3sets}
\end{document}

• Use the Venn.tex to create a PDF that displays the following

  • \( A \cap B \cup C \)
  • \( A \cup B \cup C \)
  • \( \neg(A \cap B \cap C) \)

• Upload the diagram to the Google Classroom

Create a Venn Diagram in yEd to represent the relationships between these standard sets:

\[ \mathbb{N}, \mathbb{P}, \mathbb{Q}, \mathbb{R}, \mathbb{W}, \mathbb{Z} \]

• In America politics, there is much talk about women making less money than men.
• But there are some wealthy women, so how bad is the problem.
• It would be good to know what characteristics contribute to income.
• Data from an on-line survey about IT usage in the work place will be used to determine which factors or combination of factors contribute most to the income level of American women

• Education and experience
• Child-bearing and nursing
• Male dominance and competitiveness
• Gender barriers and activities
• Cultural norms and traditional roles
• Female CEO tend to invest in their committee and their workers

• Which factors are most associated with the highest female income level?

• Which factors related with the lowest female income?

• Which factors have the most impact on income?

• Which pair of factors has the most impact?

1. Use Git to pull the latest copy of the software
2. Adjust the software to look at the factors as binary values.
3. Use the ruby software to obtain counts.
4. Create a 3-factor Venn diagram of the top two factors most associated with high income in women.
5. Create a 3-factor Venn diagram of the factors whose lack is associated with the lowest income.
6. Determine the scaling factor for the factors and if there is a combination of factors that synergize the likelihood of high income.
7. Submit a research report of your findings in IEEE template format using LaTeX.

The ratio of the percentage of high income individuals between subgroups represents the magnitude of the effect. Combinations of factors identify whether the combined effects are additive or synergistic.

Your task is to compare how factors or pairs of factors effect the percentage of female respondent receiving high incomes.

EDUCATION: (Highest level attained)

EDUCATION_GROUP

JOB_ROLE

REGION: (location)

A function is a rule that assigns each input exactly one output.

Domain is the set of all possible inputs.

Range is the set of all possible outputs.

Given the following mappings, which would qualitfy as functions?

Surjection where the elements of the domain cover the full range of value in the range.

\[ f : \{1, 2, 3, 4, 5, 6\} \Rightarrow \{a,b,c\} \]

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline Original\ domain & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline Outcomes\ value & a & a & b & b & b & c \\ \hline \end{array} \]

Injection where the elements of the domain map to unique values in the range (No duplicates).

\[ f : \{1, 2, 3, 4, 5\} \Rightarrow \{a,b,c, d, e, f\} \]

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline Original\ domain & 1 & 2 & 3 & 4 & 5 \\ \hline Outcomes\ value & e & c & b & d & a \\ \hline \end{array} \]

Bijection where the all elements of the domain map to unique values which cover the entire range.

\[ f : \{1, 2, 3, 4, 5, 6\} \Rightarrow \{a,b,c, d, e, f\} \]

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline Original\ domain & 1 & 2 & 3 & 4 & 5 & 6\\ \hline Outcomes\ value & f & c & b & d & a & e\\ \hline \end{array} \]

• In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place?

• If there are 10 people and 3 chairs, how many groups of 3 are possible?

• How many ways can you distribute 10 cookies to 7 students?

The additive principle states that if event \( A \) can occur in \( m \) ways, and event \( B \) can occur in \( n \) disjoint ways, then the event \( A \cup B \) can occur in \( m + n \) ways.

• For 2 sets: \( |A \cup B| = |A| + |B| - |A \cap B| \)

• For 3 sets: \( \small|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| +|A \cap B \cap C| \)

For given sets \( A \) and \( B \), the set of all ordered pairs \( \{x,y\} \) where \( x \in A \) and \( y \in B \)

\[ \hbox{For}\ A = \{1,2,3\}\ \hbox{and}\ B = \{A,B,C,D\} \]

\[ A \times B= \left\{ \begin{array}{ccc} \{1,A\} & \{2,A\} & \{3,A\}\\ \{1,B\} & \{2,B\} & \{3,B\}\\ \{1,C\} & \{2,C\} & \{3,C\}\\ \{1,D\} & \{2,D\} & \{3,D\}\\ \end{array} \right\} \]

If event \( A \) can occur in \( m \) ways, and each possibility for \( A \) allows for exactly \( n \) ways for event \( B \), then the event \( A \cup B \) can occur in \( m \times n \) ways

For sets: \( |A \times B| = |A| \cdot |B| \)

\[ \small\begin{array}{cccc} \textbf{n} & \textbf{Sequence} &\textbf{Sum} &\textbf{Equiv}\\ \hline 1 & \small 1 & 1 & 1{(1+1)\over 2} \\ 2 & \small 1+2 & 3 & 2{(1+2)\over 2} \\ 3 & \small 1+2+3 & 6 & 3{(1+3)\over 2} \\ 4 & \small 1+2+3+4 & 10 & 4{(1+4)\over 2} \\ 5 & \small 1+2+3+4+5 & 15 & 5{(1+5)\over 2} \\ 6 & \small 1+2+3+4+5+6 & 21 & 6{(1+6)\over 2} \\ 7 & \small 1+2+3+4+5+6+7 & 28 & 7{(1+7)\over 2} \\ 8 & \small 1+2+3+4+5+6+7+8 & 36 & 8{(1+8)\over 2} \\ \hline \end{array} \]

\[ \begin{array}{rcc} count & = & \sum{\{1 .. n\}} + \sum{\{1 .. (n-1)\}} + ... + \{1\} \\ & = & {x(x+1)\over 2} + {(x-1)((x-1)+1)\over 2} + ... + 1 \\ & = & {x_n + x_n^2\over 2} + {x_{n-1} + x_{n-1}^2\over 2} + ... + {x_{1} + x_{1}^2\over 2}\\ & = & {\sum_{i=1}^{n}\ x_i + \sum_{i=1}^{n}\ x_i^2\over 2} \\ \end{array} \]

If \( N \) objects are placed in \( K \) boxes, then there is at least one box containing at least \( \lceil N/K \rceil \) objects.

Example:

Among 100 people, there are at least \( \lceil 100 / 12 \rceil = 9 \) who were born in the same month.

• A drawer contains a dozen brown socks and a dozen black sock, all unmatched. A man takes socks out at random in the dark.

  • How many socks must he take out to be sure that he has at least 2 socks of the same Color?
  • How many socks must he take out to be sure that he has at least two black socks?

• If there are five possible grades (i.e., A, B, C, D, F), what is the minimum number of students needed to Ensure that at least six students get the same grade.

• A palindrome is a string that is identical to the string in reverse order. How many bit strings of length \( n \) are palindromes?

• How many licence plates can be made using either two letters followed by 4 digits or four letters followed by 2 digit is?

• How many books can be identified using the 12 digit ISBN?

• Total number of subsets:

\[ |{\cal P}(A)| = 2^5 = 32 \]

• Total number of subsets of size \( n \) :

\[ \left({|A|\over n!\ (|A| - n)!}\right) \]

\[ \small\begin{array}{|c|c|c|c|c|} \hline n=0 & n=1 & n=2 & n=3 & n=4 & n=5 \\ \hline \left({5\atop 0}\right) & \left({5\atop 1}\right) & \left({5\atop 2}\right) & \left({5\atop 3}\right) & \left({5\atop 4}\right) & \left({5\atop 5}\right) \\ \hline \left({5!\over 0!\ 5!}\right) & \left({5!\over 1!\ 4!}\right) & \left({5!\over 2!\ 3!}\right) & \left({5!\over 3!\ 2!}\right) & \left({5!\over 4!\ 1!}\right) & \left({5!\over 5!\ 0!}\right) \\ \hline {5\cdot4\cdot3\cdot2\cdot1\over (5\cdot4\cdot3\cdot2\cdot1)} & {5\cdot4\cdot3\cdot2\cdot1\over (1)(4\cdot3\cdot2\cdot1)} & {5\cdot4\cdot3\cdot2\cdot1\over (2\cdot1)(3\cdot2\cdot1)} & {5\cdot4\cdot3\cdot2\cdot1\over (3\cdot2\cdot1)(2\cdot1)} & {5\cdot4\cdot3\cdot2\cdot1\over (4\cdot3\cdot2\cdot1)(1)} & {5\cdot4\cdot3\cdot2\cdot1\over (5\cdot4\cdot3\cdot2\cdot1)} \\ \hline {1 \over 1} & {5 \over 1} & {5\cdot4 \over 2\cdot1} = {5\cdot2\over 1} & {5\cdot4\cdot3 \over 3\cdot2\cdot1} = {5\cdot 2\over 1} & {5 \over 1} & {1 \over 1}\\ \hline 1 & 5 & 10 & 10 & 5 & 1 \\ \hline \emptyset & \{1\}, & \{1,2\}, \{2,3\}, &\{1,2,3\}, \{2,3,4\}, & \{1,2,3,4\}, & \{1,2,3,4,5\}\\ & \{2\}, &\{3,4\}, \{4,5\},& \{3,4,5\}, \{1,2,4\}, & \{1,2,3,5\}, & \\ &\{3\}, & \{1,3\}, \{1,4\}, &\{2,3,5\}, \{1,2,5\}, & \{1,2,4,5\}, & \\ &\{4\}, &\{1,5\}, \{1,4\}, &\{1,3,4\}, \{1,3,5\},&\{1,3,4,5\},&\\ &\{5\} &\{2,5\}, \{1,5\} & \{1,4,5\}, \{2,4,5\} & \{2,3,4,5\} & \\ \hline 00000 & 10000, & 11000,01100, & 11100, 01110, & 11110, & 11111 \\ & 01000, & 00110, 00011, & 00111, 11010, & 11101, & \\ & 00100, & 10100, 10010, & 01101, 11001, & 11011, & \\ & 00010, & 10001, 10010, & 10110, 10101, & 10111, &\\ & 00001 & 01001, 10001 & 10011, 01011 & 01111 & \\ \hline \end{array} \]

• An \( n \) - bit string is a bit string of length n. That is, it is a string containing \( n \) symbols, each of which is a bit, either 0 or 1.
• The weight of a bit string is the number of 1’s in it.
• \( B^n \) is the set of all \( n \) - bit strings, \( B^n_k \) is the set of all \( n \) - bit strings of weight \( k \) .

\[ \small\begin{array}{|l|c|c|c|c|c|} \hline weight: & k=0 & k=1 & k=2 & k=3 & k=4 & k=5 \\ \hline B^5_k & 00000 & 10000, & 11000, 01100, & 11100, 01110, & 11110,& 11111 \\ & & 01000, & 00110, 00011, & 00111, 10110, & 11101, & \\ & & 00100, & 10100, 01010, & 01011, 10011, & 11011, & \\ & & 00010, & 00101, 10010, & 10101, 11001, & 10111, & \\ & & 00001 & 01001, 10001 & 11010, 01101 & 01111 & \\ \hline B^5_k& 1 & 5 & 10 & 10 & 5 & 1 \\ \hline \end{array} \]

• Recursively break the problem down by imposing a value of the first digit of the unknown set and solving the problem for the resulting subsets:

  \[ \{?????\}_{k=3} = 0\{????\}_{k=3} + 1\{????\}_{k=2} \]

• Once the problem has been reduced to a simple subset, the values are back substituted.

\[ \begin{array}{rcccccl} |B^5_3| & = & |B^4_3| + |B^4_2| &=& 4 + 6 & = & 10 \\ |B^4_3| & = & |B^3_3| + |B^3_2| &=& 1 + 3 & = & 4 \\ |B^4_2| & = & |B^3_2| + |B^3_1| &=& 3 + 3 & = & 6 \\ \end{array} \]

\[ \begin{array}{c} (x + y)^1 = x + y\\ (x + y)^2 = x^2 + 2xy + y^2\\ (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\\ (x + y)^4 = x^4 + +4x^3y + 6x^2y^2 + 4xy^3 + y^4\\ (x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5\\ \end{array} \]

\[ \begin{array}{c} \exists n,k: n \ge 0, 0 \le k \le n \Rightarrow\\ \left({n \atop k}\right)\\ \end{array} \]

• Read as n choose k.
• Equals \( |B^n_k| \) the number of \( n \) - bit strings of weight \( k \)
• Equals the number of subsets of size \( n \) with cardinality \( k \)
• Equals the number of lattice paths of length \( n \) with \( k \) steps to the right.
• Equals the coefficient of \( x^k y^{n−k} \) in the expansion of \( (x + y)^n \)
• Equals the number of ways to select \( k \) objects from a total of \( n \) objects

\[ \left({n \atop k}\right) =\left({n-1 \atop k-1}\right) + \left({n-1 \atop k}\right) \]