garywelz/programming_framework
0
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 