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1<!DOCTYPE html>2<html lang="en">3<head>4    <meta charset="UTF-8" />5    <meta name="viewport" content="width=device-width, initial-scale=1" />6    <title>Mathematics Batch 06 - Discrete Mathematics - Programming Framework Analysis</title>7    <style>8        body { 9            font-family: 'Times New Roman', Times, serif, 'Arial Unicode MS'; 10            margin: 0; 11            background: #ffffff; 12            color: #000000; 13            line-height: 1.6; 14            font-size: 12pt; 15        }16        .container { 17            max-width: 1000px; 18            margin: 0 auto; 19            padding: 1.5rem; 20        }21        h1, h2, h3 { 22            color: #000000; 23            margin-top: 1.5rem; 24            margin-bottom: 0.75rem; 25        }26        h1 { 27            font-size: 18pt; 28            text-align: center; 29        }30        h2 { 31            font-size: 16pt; 32            border-bottom: 2px solid #000; 33            padding-bottom: 0.5rem; 34        }35        h3 { 36            font-size: 14pt; 37        }38        p { 39            margin-bottom: 1rem; 40            text-align: justify; 41        }42        .figure { 43            margin: 2rem 0; 44            text-align: center; 45            border: 1px solid #ccc; 46            padding: 1rem; 47            background: #f9f9f9; 48        }49        .figure-caption { 50            margin-top: 1rem; 51            font-style: italic; 52            text-align: left; 53        }54        .mermaid { 55            background: white; 56            padding: 1rem; 57            border-radius: 4px; 58        }59        .navigation {60            margin: 3rem 0;61            padding: 1rem;62            background: #f8f9fa;63            border-radius: 8px;64        }65        .nav-links {66            display: flex;67            flex-wrap: wrap;68            gap: 1rem;69            justify-content: center;70        }71        .nav-link {72            color: #007bff;73            text-decoration: none;74            padding: 0.5rem 1rem;75            border: 1px solid #007bff;76            border-radius: 4px;77            transition: all 0.3s ease;78        }79        .nav-link:hover {80            background: #007bff;81            color: white;82        }83        .footer {84            margin-top: 3rem;85            padding: 1rem;86            background: #f8f9fa;87            border-radius: 8px;88            text-align: center;89        }90        .contact-info {91            margin-top: 1rem;92        }93        .contact-info p {94            margin: 0.25rem 0;95            text-align: center;96        }97    </style>98    <script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script>99    <script>100        mermaid.initialize({ 101            startOnLoad: true, 102            theme: 'default', 103            flowchart: { 104                useMaxWidth: false, 105                htmlLabels: true,106                curve: 'linear',107                nodeSpacing: 30,108                rankSpacing: 30,109                padding: 10110            },111            themeVariables: {112                fontFamily: 'Arial Unicode MS, Arial, sans-serif'113            }114        });115    </script>116</head>117<body>118    <div class="container">119        <h1>Mathematics Batch 06 - Discrete Mathematics - Programming Framework Analysis</h1>120        121        <p>This document presents discrete mathematics processes analyzed using the Programming Framework methodology. Each process is represented as a computational flowchart with standardized color coding: Red for triggers/inputs, Yellow for structures/objects, Green for processing/operations, Blue for intermediates/states, and Violet for products/outputs. Yellow nodes use black text for optimal readability, while all other colors use white text.</p>122 123        <h2>1. Graph Theory Algorithms Process</h2>124            <div class="figure">125                <div class="mermaid">126graph TD127    A1[Graph G V E] --> B1[Graph Representation]128    C1[Algorithm Selection] --> D1[Method Choice]129    E1[Graph Properties] --> F1[Property Analysis]130    131    B1 --> G1[Adjacency Matrix]132    D1 --> H1[Shortest Path Algorithm]133    F1 --> I1[Graph Connectivity]134    135    G1 --> J1[Adjacency List]136    H1 --> K1[Dijkstra Algorithm]137    I1 --> L1[Graph Traversal]138    139    J1 --> M1[Graph Structure]140    K1 --> N1[Bellman Ford Algorithm]141    L1 --> O1[Depth First Search]142    143    M1 --> P1[Vertex Analysis]144    N1 --> O1145    O1 --> Q1[Breadth First Search]146    147    P1 --> R1[Edge Analysis]148    Q1 --> S1[Minimum Spanning Tree]149    R1 --> T1[Graph Coloring]150    151    S1 --> U1[Kruskal Algorithm]152    T1 --> V1[Network Flow]153    U1 --> W1[Graph Algorithm Result]154    155    V1 --> X1[Ford Fulkerson Algorithm]156    W1 --> Y1[Graph Analysis]157    X1 --> Z1[Graph Theory Output]158    159    Y1 --> AA1[Graph Theory Analysis]160    Z1 --> BB1[Graph Theory Final Result]161    AA1 --> CC1[Graph Theory Analysis Complete]162    163    style A1 fill:#ff6b6b,color:#fff164    style C1 fill:#ff6b6b,color:#fff165    style E1 fill:#ff6b6b,color:#fff166    167    style B1 fill:#ffd43b,color:#000168    style D1 fill:#ffd43b,color:#000169    style F1 fill:#ffd43b,color:#000170    style G1 fill:#ffd43b,color:#000171    style H1 fill:#ffd43b,color:#000172    style I1 fill:#ffd43b,color:#000173    style J1 fill:#ffd43b,color:#000174    style K1 fill:#ffd43b,color:#000175    style L1 fill:#ffd43b,color:#000176    style M1 fill:#ffd43b,color:#000177    style N1 fill:#ffd43b,color:#000178    style O1 fill:#ffd43b,color:#000179    style P1 fill:#ffd43b,color:#000180    style Q1 fill:#ffd43b,color:#000181    style R1 fill:#ffd43b,color:#000182    style S1 fill:#ffd43b,color:#000183    style T1 fill:#ffd43b,color:#000184    style U1 fill:#ffd43b,color:#000185    style V1 fill:#ffd43b,color:#000186    style W1 fill:#ffd43b,color:#000187    style X1 fill:#ffd43b,color:#000188    style Y1 fill:#ffd43b,color:#000189    style Z1 fill:#ffd43b,color:#000190    style AA1 fill:#ffd43b,color:#000191    style BB1 fill:#ffd43b,color:#000192    style CC1 fill:#ffd43b,color:#000193    194    style M1 fill:#51cf66,color:#fff195    style N1 fill:#51cf66,color:#fff196    style O1 fill:#51cf66,color:#fff197    style P1 fill:#51cf66,color:#fff198    style Q1 fill:#51cf66,color:#fff199    style R1 fill:#51cf66,color:#fff200    style S1 fill:#51cf66,color:#fff201    style T1 fill:#51cf66,color:#fff202    style U1 fill:#51cf66,color:#fff203    style V1 fill:#51cf66,color:#fff204    style W1 fill:#51cf66,color:#fff205    style X1 fill:#51cf66,color:#fff206    style Y1 fill:#51cf66,color:#fff207    style Z1 fill:#51cf66,color:#fff208    style AA1 fill:#51cf66,color:#fff209    style BB1 fill:#51cf66,color:#fff210    style CC1 fill:#51cf66,color:#fff211    212    style CC1 fill:#b197fc,color:#fff213                </div>214                215                <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">216                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">217                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs218                    </div>219                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">220                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Graph Theory Methods221                    </div>222                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">223                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Graph Operations224                    </div>225                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">226                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates227                    </div>228                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">229                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products230                    </div>231                </div>232                233                <div class="figure-caption">234                    <strong>Figure 1.</strong> Graph Theory Algorithms Process. This discrete mathematics process visualization demonstrates shortest path, minimum spanning tree, and network flow algorithms. The flowchart shows graph inputs and properties, graph theory methods and algorithms, graph operations and calculations, intermediate results, and final graph theory outputs.235                </div>236            </div>237        </div>238 239        <h2>2. Combinatorics Process</h2>240            <div class="figure">241                <div class="mermaid">242graph TD243    A2[Combinatorial Problem] --> B2[Counting Method Selection]244    C2[Permutation Analysis] --> D2[Combination Analysis]245    E2[Counting Principles] --> F2[Principle Application]246    247    B2 --> G2[Multiplication Principle]248    D2 --> H2[Combination Formula]249    F2 --> I2[Addition Principle]250    251    G2 --> J2[Permutation Formula]252    H2 --> K2[Binomial Coefficient]253    I2 --> L2[Inclusion Exclusion]254    255    J2 --> M2[Factorial Calculation]256    K2 --> N2[Pascal Triangle]257    L2 --> O2[Set Cardinality]258    259    M2 --> P2[Arrangement Counting]260    N2 --> O2261    O2 --> Q2[Partition Counting]262    263    P2 --> R2[Selection Counting]264    Q2 --> S2[Generating Functions]265    R2 --> T2[Combinatorial Identity]266    267    S2 --> U2[Recurrence Relations]268    T2 --> V2[Combinatorial Proof]269    U2 --> W2[Combinatorial Result]270    271    V2 --> X2[Combinatorial Analysis]272    W2 --> Y2[Combinatorial Output]273    X2 --> Z2[Combinatorial Analysis Output]274    275    Y2 --> AA2[Combinatorial Analysis Final Result]276    Z2 --> BB2[Combinatorial Analysis Complete]277    AA2 --> CC2[Combinatorial Analysis Output]278    279    style A2 fill:#ff6b6b,color:#fff280    style C2 fill:#ff6b6b,color:#fff281    style E2 fill:#ff6b6b,color:#fff282    283    style B2 fill:#ffd43b,color:#000284    style D2 fill:#ffd43b,color:#000285    style F2 fill:#ffd43b,color:#000286    style G2 fill:#ffd43b,color:#000287    style H2 fill:#ffd43b,color:#000288    style I2 fill:#ffd43b,color:#000289    style J2 fill:#ffd43b,color:#000290    style K2 fill:#ffd43b,color:#000291    style L2 fill:#ffd43b,color:#000292    style M2 fill:#ffd43b,color:#000293    style N2 fill:#ffd43b,color:#000294    style O2 fill:#ffd43b,color:#000295    style P2 fill:#ffd43b,color:#000296    style Q2 fill:#ffd43b,color:#000297    style R2 fill:#ffd43b,color:#000298    style S2 fill:#ffd43b,color:#000299    style T2 fill:#ffd43b,color:#000300    style U2 fill:#ffd43b,color:#000301    style V2 fill:#ffd43b,color:#000302    style W2 fill:#ffd43b,color:#000303    style X2 fill:#ffd43b,color:#000304    style Y2 fill:#ffd43b,color:#000305    style Z2 fill:#ffd43b,color:#000306    style AA2 fill:#ffd43b,color:#000307    style BB2 fill:#ffd43b,color:#000308    style CC2 fill:#ffd43b,color:#000309    310    style M2 fill:#51cf66,color:#fff311    style N2 fill:#51cf66,color:#fff312    style O2 fill:#51cf66,color:#fff313    style P2 fill:#51cf66,color:#fff314    style Q2 fill:#51cf66,color:#fff315    style R2 fill:#51cf66,color:#fff316    style S2 fill:#51cf66,color:#fff317    style T2 fill:#51cf66,color:#fff318    style U2 fill:#51cf66,color:#fff319    style V2 fill:#51cf66,color:#fff320    style W2 fill:#51cf66,color:#fff321    style X2 fill:#51cf66,color:#fff322    style Y2 fill:#51cf66,color:#fff323    style Z2 fill:#51cf66,color:#fff324    style AA2 fill:#51cf66,color:#fff325    style BB2 fill:#51cf66,color:#fff326    style CC2 fill:#51cf66,color:#fff327    328    style CC2 fill:#b197fc,color:#fff329                </div>330                331                <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">332                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">333                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs334                    </div>335                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">336                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Combinatorial Methods337                    </div>338                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">339                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Counting Operations340                    </div>341                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">342                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates343                    </div>344                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">345                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products346                    </div>347                </div>348                349                <div class="figure-caption">350                    <strong>Figure 2.</strong> Combinatorics Process. This discrete mathematics process visualization demonstrates permutations, combinations, and counting principles. The flowchart shows combinatorial problem inputs and principles, combinatorial methods and formulas, counting operations and calculations, intermediate results, and final combinatorial analysis outputs.351                </div>352            </div>353        </div>354 355        <h2>3. Logic & Set Theory Process</h2>356            <div class="figure">357                <div class="mermaid">358graph TD359    A3[Logical Statement Input] --> B3[Logical Analysis]360    C3[Set Operations] --> D3[Set Theory Application]361    E3[Proof Techniques] --> F3[Proof Strategy]362    363    B3 --> G3[Boolean Algebra]364    D3 --> H3[Set Union Intersection]365    F3 --> I3[Direct Proof]366    367    G3 --> J3[Logical Connectives]368    H3 --> K3[Set Complement]369    I3 --> L3[Contradiction Proof]370    371    J3 --> M3[Truth Tables]372    K3 --> L3373    L3 --> N3[Induction Proof]374    375    M3 --> O3[Logical Equivalence]376    N3 --> P3[Proof by Cases]377    O3 --> Q3[Set Cardinality]378    379    P3 --> R3[Proof Validation]380    Q3 --> S3[Set Operations Result]381    R3 --> T3[Logical Result]382    383    S3 --> U3[Set Theory Analysis]384    T3 --> V3[Logical Analysis]385    U3 --> W3[Logic Set Theory Result]386    387    V3 --> X3[Logic Set Theory Analysis]388    W3 --> Y3[Logic Set Theory Output]389    X3 --> Z3[Logic Set Theory Analysis Output]390    391    Y3 --> AA3[Logic Set Theory Analysis Final Result]392    Z3 --> BB3[Logic Set Theory Analysis Complete]393    AA3 --> CC3[Logic Set Theory Analysis Output]394    395    style A3 fill:#ff6b6b,color:#fff396    style C3 fill:#ff6b6b,color:#fff397    style E3 fill:#ff6b6b,color:#fff398    399    style B3 fill:#ffd43b,color:#000400    style D3 fill:#ffd43b,color:#000401    style F3 fill:#ffd43b,color:#000402    style G3 fill:#ffd43b,color:#000403    style H3 fill:#ffd43b,color:#000404    style I3 fill:#ffd43b,color:#000405    style J3 fill:#ffd43b,color:#000406    style K3 fill:#ffd43b,color:#000407    style L3 fill:#ffd43b,color:#000408    style M3 fill:#ffd43b,color:#000409    style N3 fill:#ffd43b,color:#000410    style O3 fill:#ffd43b,color:#000411    style P3 fill:#ffd43b,color:#000412    style Q3 fill:#ffd43b,color:#000413    style R3 fill:#ffd43b,color:#000414    style S3 fill:#ffd43b,color:#000415    style T3 fill:#ffd43b,color:#000416    style U3 fill:#ffd43b,color:#000417    style V3 fill:#ffd43b,color:#000418    style W3 fill:#ffd43b,color:#000419    style X3 fill:#ffd43b,color:#000420    style Y3 fill:#ffd43b,color:#000421    style Z3 fill:#ffd43b,color:#000422    style AA3 fill:#ffd43b,color:#000423    style BB3 fill:#ffd43b,color:#000424    style CC3 fill:#ffd43b,color:#000425    426    style M3 fill:#51cf66,color:#fff427    style N3 fill:#51cf66,color:#fff428    style O3 fill:#51cf66,color:#fff429    style P3 fill:#51cf66,color:#fff430    style Q3 fill:#51cf66,color:#fff431    style R3 fill:#51cf66,color:#fff432    style S3 fill:#51cf66,color:#fff433    style T3 fill:#51cf66,color:#fff434    style U3 fill:#51cf66,color:#fff435    style V3 fill:#51cf66,color:#fff436    style W3 fill:#51cf66,color:#fff437    style X3 fill:#51cf66,color:#fff438    style Y3 fill:#51cf66,color:#fff439    style Z3 fill:#51cf66,color:#fff440    style AA3 fill:#51cf66,color:#fff441    style BB3 fill:#51cf66,color:#fff442    style CC3 fill:#51cf66,color:#fff443    444    style CC3 fill:#b197fc,color:#fff445                </div>446                447                <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">448                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">449                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs450                    </div>451                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">452                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Logic Set Methods453                    </div>454                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">455                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Proof Operations456                    </div>457                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">458                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates459                    </div>460                    <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">461                        <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products462                    </div>463                </div>464                465                <div class="figure-caption">466                    <strong>Figure 3.</strong> Logic & Set Theory Process. This discrete mathematics process visualization demonstrates boolean algebra, set operations, and proof techniques. The flowchart shows logical statement inputs and proof techniques, logic and set theory methods and algorithms, proof operations and calculations, intermediate results, and final logic and set theory analysis outputs.467                </div>468            </div>469        </div>470 471        <div class="navigation">472            <h3>Navigation</h3>473            <div class="nav-links">474                <a href="mathematics_index.html" class="nav-link">← Back to Mathematics Index</a>475                <a href="mathematics_batch_05.html" class="nav-link">← Previous: Applied Mathematics</a>476                <a href="mathematics_batch_07.html" class="nav-link">Next: Historical & Educational →</a>477                <a href="index.html" class="nav-link">Programming Framework Home</a>478            </div>479        </div>480 481        <div class="footer">482            <p><strong>Generated using the Programming Framework methodology</strong></p>483            <p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p>484            <div class="contact-info">485                <p><strong>Gary Welz</strong></p>486                <p>Retired Faculty Member</p>487                <p>John Jay College, CUNY (Department of Mathematics and Computer Science)</p>488                <p>Borough of Manhattan Community College, CUNY</p>489                <p>CUNY Graduate Center (New Media Lab)</p>490                <p>Email: gwelz@jjay.cuny.edu</p>491            </div>492        </div>493    </div>494</body>495</html>496