I’m going to attempt to explain relations and their different properties. This was a project in my discrete math class that I believe can help anyone to understand what relations are. Before I explain the code, here are the basic properties of relations with examples. In each example R is the given relation.
Reflexive – R is reflexive if every element relates to itself. {(1,1) (2,2)(3,3)}
Irreflexive – R is irreflexive if every element does not relate to itself. {(1,2) (1,3) (2,1) (2,3) (3,1) (3,2)}
Symmetric – R is symmetric if a relates to b (a->b), then b relates to a (b->a). {(1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2)}
Antisymmetric – R is antisymmetric if a relates to b (a->b), and b relates to a (b->a), then a must equal b (a = b). {(3,2) (3,3)}
– is antisymmetric
– is not antisymmetric
Asymmetric – if a relates to b (a->b), then b does not relate to a (b!->a). {(1,2) (3,1) (3,2)}
Transitive – if a relates to b (a->b), and b relates to c (b->c), then a relates to c (a->c).
– is transitive
– is not transitive
Now that we understand the properties we can talk about the code. The code takes in one argument from the command line that is the file with the Relation. The file needs to contain the relation in a matrix form like the examples above with the first number the size of the matrix. Here is an example:
The file above would be the following relation: {(1,2) (2,3)}. There are spaces to separate each cell of the matrix. After given a relation the program will output which properties hold and which ones don’t. We can study the source code to see how to test for each condition. I do not claim this program to be the most efficient way to determine the different relation properties. Please leave in the comments any questions or ideas that you might have.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 | /*************************************************************************** * Program: * Relations as Connection Matrices * Author: * Don Page * Summary: * Represents relations as connection (zero-one) matrices, and provides * functionality for testing properties of relations. * ***************************************************************************/ #include <cmath> #include <iostream> #include <fstream> #include <iomanip> #include <assert.h> using namespace std; class Relation { private: bool** mMatrix; int mSize; void init() { mMatrix = new bool*[mSize]; for (int i = 0; i < mSize; i++) { mMatrix[i] = new bool[mSize]; } } public: Relation(int size) { mSize = size; init(); } Relation& operator=(const Relation& rtSide) { if (this == &rtSide) { return *this; } else { mSize = rtSide.mSize; for (int i = 0; i < mSize; i++) { delete [] mMatrix[i]; } delete [] mMatrix; init(); for (int x = 0; x < mSize; x++) { for (int y = 0; y < mSize; y++) { mMatrix[x][y] = rtSide[x][y]; } } } return *this; } Relation(const Relation& relation) { mSize = relation.getConnectionMatrixSize(); init(); *this = relation; } ~Relation() { for (int i = 0; i < mSize; i++) { delete [] mMatrix[i]; } delete [] mMatrix; } bool isReflexive(); bool isIrreflexive(); bool isNonreflexive(); bool isSymmetric(); bool isAntisymmetric(); bool isAsymmetric(); bool isTransitive(); void describe(); int getConnectionMatrixSize() const { return mSize; } bool* operator[](int row) const { return mMatrix[row]; } bool operator==(const Relation& relation) { int size = relation.getConnectionMatrixSize(); if (mSize != size) { return false; } for (int i = 0; i < size; i++) { for (int j = 0; j < size; j++) { if (mMatrix[i][j] != relation[i][j]) { return false; } } } return true; } /**************************************************************************** * Returns product of 2 square matrices. Algorithm used from Rosen's Discrete * Mathematics and Its Applications p.253 ***************************************************************************/ Relation operator * (const Relation& relation) { // assume multiplying square matrices assert(mSize == relation.getConnectionMatrixSize()); Relation product(mSize); for (int i = 0; i < mSize; i++) { for (int j = 0; j < mSize; j++) { product.mMatrix[i][j] = 0; for (int k = 0; k < mSize; k++) { product.mMatrix[i][j] = product.mMatrix[i][j] || (mMatrix[i][k] && relation.mMatrix[k][j]); } } } return product; } /**************************************************************************** * Matrix A is less than Matrix B iff there is a 1 in B everywhere there * is a 1 in A ***************************************************************************/ bool operator <= (const Relation& relation) { for (int i = 0; i < mSize; i++) { for (int j = 0; j < mSize; j++) { if (mMatrix[i][j] && !relation.mMatrix[i][j]) return false; } } return true; } }; ostream& operator<<(ostream& os, const Relation& relation) { int n = relation.getConnectionMatrixSize(); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { os << relation[i][j] << " "; } os << endl; } return os; } istream& operator>>(istream& is, Relation& relation) { int n = relation.getConnectionMatrixSize(); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { is >> relation[i][j]; } } return is; } /**************************************************************************** * Relation Member Functions ***************************************************************************/ /**************************************************************************** * R is Reflexive if M[i][i] = 1 for all i ***************************************************************************/ bool Relation::isReflexive() { for (int i = 0; i < mSize; i++) { if (!mMatrix[i][i]) return false; } return true; } /**************************************************************************** * R is Irreflexive if M[i][i] = 0 for all i ***************************************************************************/ bool Relation::isIrreflexive() { for (int i = 0; i < mSize; i++) { if (mMatrix[i][i]) return false; } return true; } /**************************************************************************** * R is Nonreflexive if R is either not reflexive or not irreflexive. There * is a more efficient way to test for nonreflexive, but for learning purposes * I chose this inefficient way. ***************************************************************************/ bool Relation::isNonreflexive() { return (!(isReflexive() || isIrreflexive())); } /**************************************************************************** * R is Symmetric if for each M[i][j] = 1 , M[j][i] = 1 for all i,j ***************************************************************************/ bool Relation::isSymmetric() { for (int x = 0; x < mSize; x++) { for (int y = 0; y < mSize; y++) { if (mMatrix[x][y] && !mMatrix[y][x]) return false; } } return true; } /**************************************************************************** * R is AntiSymmetric if for each M[i][j] = 1 and M[j][i] = 1, then i = j * for all i,j ***************************************************************************/ bool Relation::isAntisymmetric() { for (int x = 0; x < mSize; x++) { for (int y = 0; y < mSize; y++) { if (mMatrix[x][y] && mMatrix[y][x] && (x != y)) return false; } } return true; } /**************************************************************************** * R is Asymmetric if M[i][j] = 1, then M[j][i] != 1 for all i,j ***************************************************************************/ bool Relation::isAsymmetric() { for (int x = 0; x < mSize; x++) { for (int y = 0; y < mSize; y++) { if (mMatrix[x][y] && mMatrix[y][x]) return false; } } return true; } /**************************************************************************** * R is Transitive if R^2 <= R. Another way to test if a relation is * transitive would be if M[i][j] = 1, and M[j][k] = 1, then M[i][k] = 1. * This would require 3 nested for loops. ***************************************************************************/ bool Relation::isTransitive() { Relation relation = *this; Relation product = relation * relation; return (product <= relation); } /**************************************************************************** * Describes the matrix after testing * Reflextivity, Irreflextivity, NonReflextivity, Symmetry, AntiSymmetry, * Asymmetry, and Transitivity ***************************************************************************/ void Relation::describe() { cout << "\nThe relation represented by the " << mSize << "x" << mSize << " matrix\n"; cout << *this << "is\n"; cout << (isReflexive() ? "" : "NOT ") << "Reflexive\n"; cout << (isIrreflexive() ? "" : "NOT ") << "Irreflexive\n"; cout << (isNonreflexive() ? "" : "NOT ") << "Nonreflexive\n"; cout << (isSymmetric() ? "" : "NOT ") << "Symmetric\n"; cout << (isAntisymmetric() ? "" : "NOT ") << "Antisymmetric\n"; cout << (isAsymmetric() ? "" : "NOT ") << "Asymmetric \n"; cout << (isTransitive() ? "" : "NOT ") << "Transitive.\n"; } int main(int argc, char* argv[]) { for (int i = 1; i < argc; i++) { string file = argv[i]; ifstream inFile(file.c_str()); if (inFile.is_open()) { int size; inFile >> size; Relation relation(size); inFile >> relation; inFile.close(); relation.describe(); } else { cout << "Unable to open " + file; } } return 0; } |