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<title>Mathematics Batch 06 - Discrete Mathematics - Programming Framework Analysis</title>
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<h1>Mathematics Batch 06 - Discrete Mathematics - Programming Framework Analysis</h1>
<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>
<h2>1. Graph Theory Algorithms Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
A1[Graph G V E] --> B1[Graph Representation]
C1[Algorithm Selection] --> D1[Method Choice]
E1[Graph Properties] --> F1[Property Analysis]
B1 --> G1[Adjacency Matrix]
D1 --> H1[Shortest Path Algorithm]
F1 --> I1[Graph Connectivity]
G1 --> J1[Adjacency List]
H1 --> K1[Dijkstra Algorithm]
I1 --> L1[Graph Traversal]
J1 --> M1[Graph Structure]
K1 --> N1[Bellman Ford Algorithm]
L1 --> O1[Depth First Search]
M1 --> P1[Vertex Analysis]
N1 --> O1
O1 --> Q1[Breadth First Search]
P1 --> R1[Edge Analysis]
Q1 --> S1[Minimum Spanning Tree]
R1 --> T1[Graph Coloring]
S1 --> U1[Kruskal Algorithm]
T1 --> V1[Network Flow]
U1 --> W1[Graph Algorithm Result]
V1 --> X1[Ford Fulkerson Algorithm]
W1 --> Y1[Graph Analysis]
X1 --> Z1[Graph Theory Output]
Y1 --> AA1[Graph Theory Analysis]
Z1 --> BB1[Graph Theory Final Result]
AA1 --> CC1[Graph Theory Analysis Complete]
style A1 fill:#ff6b6b,color:#fff
style C1 fill:#ff6b6b,color:#fff
style E1 fill:#ff6b6b,color:#fff
style B1 fill:#ffd43b,color:#000
style D1 fill:#ffd43b,color:#000
style F1 fill:#ffd43b,color:#000
style G1 fill:#ffd43b,color:#000
style H1 fill:#ffd43b,color:#000
style I1 fill:#ffd43b,color:#000
style J1 fill:#ffd43b,color:#000
style K1 fill:#ffd43b,color:#000
style L1 fill:#ffd43b,color:#000
style M1 fill:#ffd43b,color:#000
style N1 fill:#ffd43b,color:#000
style O1 fill:#ffd43b,color:#000
style P1 fill:#ffd43b,color:#000
style Q1 fill:#ffd43b,color:#000
style R1 fill:#ffd43b,color:#000
style S1 fill:#ffd43b,color:#000
style T1 fill:#ffd43b,color:#000
style U1 fill:#ffd43b,color:#000
style V1 fill:#ffd43b,color:#000
style W1 fill:#ffd43b,color:#000
style X1 fill:#ffd43b,color:#000
style Y1 fill:#ffd43b,color:#000
style Z1 fill:#ffd43b,color:#000
style AA1 fill:#ffd43b,color:#000
style BB1 fill:#ffd43b,color:#000
style CC1 fill:#ffd43b,color:#000
style M1 fill:#51cf66,color:#fff
style N1 fill:#51cf66,color:#fff
style O1 fill:#51cf66,color:#fff
style P1 fill:#51cf66,color:#fff
style Q1 fill:#51cf66,color:#fff
style R1 fill:#51cf66,color:#fff
style S1 fill:#51cf66,color:#fff
style T1 fill:#51cf66,color:#fff
style U1 fill:#51cf66,color:#fff
style V1 fill:#51cf66,color:#fff
style W1 fill:#51cf66,color:#fff
style X1 fill:#51cf66,color:#fff
style Y1 fill:#51cf66,color:#fff
style Z1 fill:#51cf66,color:#fff
style AA1 fill:#51cf66,color:#fff
style BB1 fill:#51cf66,color:#fff
style CC1 fill:#51cf66,color:#fff
style CC1 fill:#b197fc,color:#fff
</div>
<div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Graph Theory Methods
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Graph Operations
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products
</div>
</div>
<div class="figure-caption">
<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.
</div>
</div>
</div>
<h2>2. Combinatorics Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
A2[Combinatorial Problem] --> B2[Counting Method Selection]
C2[Permutation Analysis] --> D2[Combination Analysis]
E2[Counting Principles] --> F2[Principle Application]
B2 --> G2[Multiplication Principle]
D2 --> H2[Combination Formula]
F2 --> I2[Addition Principle]
G2 --> J2[Permutation Formula]
H2 --> K2[Binomial Coefficient]
I2 --> L2[Inclusion Exclusion]
J2 --> M2[Factorial Calculation]
K2 --> N2[Pascal Triangle]
L2 --> O2[Set Cardinality]
M2 --> P2[Arrangement Counting]
N2 --> O2
O2 --> Q2[Partition Counting]
P2 --> R2[Selection Counting]
Q2 --> S2[Generating Functions]
R2 --> T2[Combinatorial Identity]
S2 --> U2[Recurrence Relations]
T2 --> V2[Combinatorial Proof]
U2 --> W2[Combinatorial Result]
V2 --> X2[Combinatorial Analysis]
W2 --> Y2[Combinatorial Output]
X2 --> Z2[Combinatorial Analysis Output]
Y2 --> AA2[Combinatorial Analysis Final Result]
Z2 --> BB2[Combinatorial Analysis Complete]
AA2 --> CC2[Combinatorial Analysis Output]
style A2 fill:#ff6b6b,color:#fff
style C2 fill:#ff6b6b,color:#fff
style E2 fill:#ff6b6b,color:#fff
style B2 fill:#ffd43b,color:#000
style D2 fill:#ffd43b,color:#000
style F2 fill:#ffd43b,color:#000
style G2 fill:#ffd43b,color:#000
style H2 fill:#ffd43b,color:#000
style I2 fill:#ffd43b,color:#000
style J2 fill:#ffd43b,color:#000
style K2 fill:#ffd43b,color:#000
style L2 fill:#ffd43b,color:#000
style M2 fill:#ffd43b,color:#000
style N2 fill:#ffd43b,color:#000
style O2 fill:#ffd43b,color:#000
style P2 fill:#ffd43b,color:#000
style Q2 fill:#ffd43b,color:#000
style R2 fill:#ffd43b,color:#000
style S2 fill:#ffd43b,color:#000
style T2 fill:#ffd43b,color:#000
style U2 fill:#ffd43b,color:#000
style V2 fill:#ffd43b,color:#000
style W2 fill:#ffd43b,color:#000
style X2 fill:#ffd43b,color:#000
style Y2 fill:#ffd43b,color:#000
style Z2 fill:#ffd43b,color:#000
style AA2 fill:#ffd43b,color:#000
style BB2 fill:#ffd43b,color:#000
style CC2 fill:#ffd43b,color:#000
style M2 fill:#51cf66,color:#fff
style N2 fill:#51cf66,color:#fff
style O2 fill:#51cf66,color:#fff
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style U2 fill:#51cf66,color:#fff
style V2 fill:#51cf66,color:#fff
style W2 fill:#51cf66,color:#fff
style X2 fill:#51cf66,color:#fff
style Y2 fill:#51cf66,color:#fff
style Z2 fill:#51cf66,color:#fff
style AA2 fill:#51cf66,color:#fff
style BB2 fill:#51cf66,color:#fff
style CC2 fill:#51cf66,color:#fff
style CC2 fill:#b197fc,color:#fff
</div>
<div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Combinatorial Methods
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Counting Operations
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products
</div>
</div>
<div class="figure-caption">
<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.
</div>
</div>
</div>
<h2>3. Logic & Set Theory Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
A3[Logical Statement Input] --> B3[Logical Analysis]
C3[Set Operations] --> D3[Set Theory Application]
E3[Proof Techniques] --> F3[Proof Strategy]
B3 --> G3[Boolean Algebra]
D3 --> H3[Set Union Intersection]
F3 --> I3[Direct Proof]
G3 --> J3[Logical Connectives]
H3 --> K3[Set Complement]
I3 --> L3[Contradiction Proof]
J3 --> M3[Truth Tables]
K3 --> L3
L3 --> N3[Induction Proof]
M3 --> O3[Logical Equivalence]
N3 --> P3[Proof by Cases]
O3 --> Q3[Set Cardinality]
P3 --> R3[Proof Validation]
Q3 --> S3[Set Operations Result]
R3 --> T3[Logical Result]
S3 --> U3[Set Theory Analysis]
T3 --> V3[Logical Analysis]
U3 --> W3[Logic Set Theory Result]
V3 --> X3[Logic Set Theory Analysis]
W3 --> Y3[Logic Set Theory Output]
X3 --> Z3[Logic Set Theory Analysis Output]
Y3 --> AA3[Logic Set Theory Analysis Final Result]
Z3 --> BB3[Logic Set Theory Analysis Complete]
AA3 --> CC3[Logic Set Theory Analysis Output]
style A3 fill:#ff6b6b,color:#fff
style C3 fill:#ff6b6b,color:#fff
style E3 fill:#ff6b6b,color:#fff
style B3 fill:#ffd43b,color:#000
style D3 fill:#ffd43b,color:#000
style F3 fill:#ffd43b,color:#000
style G3 fill:#ffd43b,color:#000
style H3 fill:#ffd43b,color:#000
style I3 fill:#ffd43b,color:#000
style J3 fill:#ffd43b,color:#000
style K3 fill:#ffd43b,color:#000
style L3 fill:#ffd43b,color:#000
style M3 fill:#ffd43b,color:#000
style N3 fill:#ffd43b,color:#000
style O3 fill:#ffd43b,color:#000
style P3 fill:#ffd43b,color:#000
style Q3 fill:#ffd43b,color:#000
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style S3 fill:#ffd43b,color:#000
style T3 fill:#ffd43b,color:#000
style U3 fill:#ffd43b,color:#000
style V3 fill:#ffd43b,color:#000
style W3 fill:#ffd43b,color:#000
style X3 fill:#ffd43b,color:#000
style Y3 fill:#ffd43b,color:#000
style Z3 fill:#ffd43b,color:#000
style AA3 fill:#ffd43b,color:#000
style BB3 fill:#ffd43b,color:#000
style CC3 fill:#ffd43b,color:#000
style M3 fill:#51cf66,color:#fff
style N3 fill:#51cf66,color:#fff
style O3 fill:#51cf66,color:#fff
style P3 fill:#51cf66,color:#fff
style Q3 fill:#51cf66,color:#fff
style R3 fill:#51cf66,color:#fff
style S3 fill:#51cf66,color:#fff
style T3 fill:#51cf66,color:#fff
style U3 fill:#51cf66,color:#fff
style V3 fill:#51cf66,color:#fff
style W3 fill:#51cf66,color:#fff
style X3 fill:#51cf66,color:#fff
style Y3 fill:#51cf66,color:#fff
style Z3 fill:#51cf66,color:#fff
style AA3 fill:#51cf66,color:#fff
style BB3 fill:#51cf66,color:#fff
style CC3 fill:#51cf66,color:#fff
style CC3 fill:#b197fc,color:#fff
</div>
<div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Logic Set Methods
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Proof Operations
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates
</div>
<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;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products
</div>
</div>
<div class="figure-caption">
<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.
</div>
</div>
</div>
<div class="navigation">
<h3>Navigation</h3>
<div class="nav-links">
<a href="mathematics_index.html" class="nav-link">← Back to Mathematics Index</a>
<a href="mathematics_batch_05.html" class="nav-link">← Previous: Applied Mathematics</a>
<a href="mathematics_batch_07.html" class="nav-link">Next: Historical & Educational →</a>
<a href="index.html" class="nav-link">Programming Framework Home</a>
</div>
</div>
<div class="footer">
<p><strong>Generated using the Programming Framework methodology</strong></p>
<p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p>
<div class="contact-info">
<p><strong>Gary Welz</strong></p>
<p>Retired Faculty Member</p>
<p>John Jay College, CUNY (Department of Mathematics and Computer Science)</p>
<p>Borough of Manhattan Community College, CUNY</p>
<p>CUNY Graduate Center (New Media Lab)</p>
<p>Email: gwelz@jjay.cuny.edu</p>
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