$R⊆A\times{B}$
$S⊆B\times{C}$
$T_{1}⊆C\times{D}$
$T_{2}⊆B\times{C}$


不满足交换律，$R\circ{S}=S\circ{R}$不成立
满足结合律，$R\circ{(S\circ{T_{1}})}=(R\circ{S})\circ{T_{1}}$
并集满足分配律，$R\circ{(S\cup{T_{2}})}=(R\circ{S})\cup{(R\circ{T_{2}})}$
交集不满足分配律，$R\circ{(S\cap{T_{2}})}⊆(R\circ{S})\cap{(R\circ{T_{2}})}$
$R\circ{I_{B}}=I_{A}\circ{R}=R$
由于复合运算满足结合律，所以同关系复合可以写成幂乘的形式，即：

$R^{0}=I_{A}$
$R^{1}=R^{1+0}=R\circ{R^{0}}=R\circ{I_{A}}=R$
$R^{2}=R\circ{R}$
……
$R^{m}\circ{R^{n}}=R^{m+n}$
$(R^{m})^{n}=R^{mn}$

我尝试着编程实现了课本上集合与关系的相关内容，如集合的逆运算，复合运算，集合上关系的性质判断与闭包运算等，基本判断方法均为定义法。

代码如下：

#include<iostream>
#include<vector>
#include<algorithm>
using namespace std;
typedef vector<vector<int>> v_v;

void PrintMatrix(const v_v& v)   //打印矩阵
{
	for (auto &i : v)
	{
		for (auto j : i)
			cout << j << " ";
		cout << endl;
	}
}

void Get_Matrix(v_v& v,int num)    //得关系矩阵
{
	vector<int> A, B;
	int a, b, n;
	if (num == 1)
	{
		cout << "请输入前域的元素个数及各元素" << endl;
		cin >> n;
		v.resize(n);     //初始化v的大小
		while (n--)
		{
			cin >> a;
			A.push_back(a);
		}
		cout << "请输入陪域的元素个数及各元素" << endl;
		cin >> n;
		for (int i = 0; i != v.size(); ++i)
			v[i].resize(n);
		while (n--)
		{
			cin >> b;
			B.push_back(b);
		}
	}
	else if (num == 2)
	{
		cout << "请输入集合的元素个数及各元素" << endl;
		cin >> n;
		v.resize(n);
		for (int i = 0; i != v.size(); ++i)
			v[i].resize(n);
		while (n--)
		{
			cin >> a;
			A.push_back(a);
		}
		B = A;
	}
	cout << "请输入关系中元素个数及各个序偶" << endl;
	cin >> n;
	while (n--)
	{
		cin >> a >> b;
		auto a_loc = find(A.begin(), A.end(), a);
		auto b_loc = find(B.begin(), B.end(), b);
		int a_index = distance(A.begin(), a_loc);      //得a,b下标 
		int b_index = distance(B.begin(), b_loc);
		v[a_index][b_index] = 1;
	}
	cout << "其关系矩阵：" << endl;
	PrintMatrix(v);
}

v_v MatrixInver(const v_v& v)     //关系逆运算
{
	int row = v.size();
	int col = v[0].size();
	v_v v_inv(col);
	for (int i = 0; i != col; ++i)
		v_inv[i].resize(row);
	for (int i = 0; i != row; ++i)
		for (int j = 0; j != col; ++j)
			v_inv[j][i] = v[i][j];
	return v_inv;
}

v_v MatrixMul(const v_v& v1, const v_v& v2)   //关系复合运算
{
	v_v v_mul;
	int m = v1.size(), n = v2.size(), p = v2[0].size();
	v_mul.resize(m);
	for (int i = 0; i != m; ++i)
		v_mul[i].resize(p);
	for (int i = 0; i != m; ++i)
		for (int j = 0; j != p; ++j)
			for (int k = 0; k != n; ++k)
				v_mul[i][j] |= v1[i][k] * v2[k][j];
	return v_mul;
}

void Reflex(const v_v& v)
{
	int count = 0;
	v_v v1(v);
	for (int i = 0; i != v.size(); ++i)
		if (v[i][i] == 1)
			++count;
	if (count == v.size())
		cout << "该关系具有自反性" << endl;
	if (count == 0)
		cout << "该关系具有反自反性" << endl;
	cout << "其自反闭包的关系矩阵为" << endl;
	for (int i = 0; i != v.size(); ++i)
		v1[i][i] = 1;
	PrintMatrix(v1);
}

void Symmetry(const v_v& v)
{
	v_v v1(v);
	bool flag1 = true, flag2 = true;
	for (int i = 0; i != v.size(); ++i)
	{
		for (int j = 0; j != v.size(); ++j)
		{
			if (i != j)
			{
				if (v[i][j] != v[j][i])
					flag1 = false;
				else
					flag2 = false;
			}
		}
	}
	if (flag1)
		cout << "该关系具有对称性" << endl;
	if(flag2)
		cout << "该关系具有反对称性" << endl;
	cout << "其对称闭包的关系矩阵为" << endl;
	auto inver = MatrixInver(v);
	for (int i = 0; i != v.size(); ++i)
	{
		for (int j = 0; j != v.size(); ++j)
		{
			v1[i][j] |= inver[i][j];
		}
	}
	PrintMatrix(v1);
}

void Trans(const v_v& v)
{
	v_v v1(v);
	bool flag = true;
	for (int i = 0; i != v.size(); ++i)              //Warshall算法求传递闭包
	{
		for (int j = 0; j != v.size(); ++j)
		{
			if (v[i][j])
			{
				for (int k = 0; k != v.size(); ++k)
				{
					if (v[j][k])
						if (v[i][k] == 0)
							flag = false;
				}
			}
		}
	}
	if (flag)
		cout << "该关系具有传递性" << endl;
	cout << "其传递闭包的关系矩阵为" << endl;
	for (int i = 0; i != v.size(); ++i)
	{
		for (int j = 0; j != v.size(); ++j)
		{
			if (v[j][i])
				for (int k = 0; k != v.size(); ++k)
					v1[j][k] |= v[i][k];
		}
	}
	PrintMatrix(v1);
}

int main()
{
	int next;
	v_v v1, v2, v_mul;
	cout << "1、集合间的关系运算\t2、集合上的关系性质判断及闭包运算" << endl;
	cin >> next;
	if (next == 1)
	{
		Get_Matrix(v1,1);
		cout << "关系运算：1、逆运算\t2、复合运算" << endl;
		cin >> next;
		if (next == 1)
		{
			cout << "其结果为" << endl;
			PrintMatrix(MatrixInver(v1));
		}
		else if (next == 2)
		{
			cout << "现请输入另一矩阵相关数据" << endl;
			Get_Matrix(v2,1);
			cout << "其结果为" << endl;
			PrintMatrix(MatrixMul(v1, v2));
		}
	}
	else if (next == 2)
	{
		Get_Matrix(v1,2);
		cout << endl;
		Reflex(v1);
		cout << endl;
		Symmetry(v1);
		cout << endl;
		Trans(v1);
		cout << endl;
	}
	system("pause");
}

 

关系的运算
集合的运算


逆运算



例

R和R-1的关系



逆关系的性质


复合运算



例



结合律



分配律


逆运算性质



’R的n次幂



性质

 

1. 结合律与同一律



 

2. 关系复合运算的结合律的证明



 

3. 关系复合运算的同一律的证明



 

4.  关系复合运算的分配律的证明



 

5. 逆运算性质定律