. . f (0,0,1) < f (0,1,0) f (0,1,0) > f (0,1,1), , .
. ., f (0,0,0) = f(1,1,1), .
. . f (0,1,1) = f (1,0,1) = f (1,1,0) / 0,1,1/, .
.
:
f1(x1,x2,x3) = C0 Å C1&X1 Å C2&X2 Å C3&X3.
Ci:
f (0,0,0) = 1 / /
0 Å 1&0 Å C2&0 ÅC3&0 = 1, .., 0 = 1.
f (1,0,0)=0 / /
0 Å C1 &1 Å C 2&0 Å C 3&0 = 0, .., 1 = 1.
f (0,1,0) = 1/ /
0 Å C1 &0 Å C 2&1 Å C 3&0 = 1, .., 2 = 0.
f (0,0,1) = 0 / /
0 Å C 1&0 Å C 2&0 Å C 3&1 = 0,.., 3 = 0.
f1(x1,x2,x3) = 1.
f f 1 :
| 1 2 3 0 0 0 0 1 0 0 1 2 0 1 0 3 0 1 1 4 1 0 0 5 1 0 1 6 1 1 0 7 1 1 1 | f(x1,x2,x3) 1 | f1(x1,x2,x3) |
. . ,
.
: 1
5. , , , , 1 : 0, 1, 2, 3,6,12
:
| x1 x2 x3 x4 | f(x1 x2 x3 x4) | |
| 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 |
, 1 :
: f(x1,x2,x3,4 ) =
: f(x1,x2,x3,4)=
, 0 :
:
f(x1,x2,x3,4)= 
:
f(x1,x2,x3,4)=
:
1) 1 1 :
f(x1,x2,x3,4) =
2) 0 0 :
f(x1,x2,x3,4)=
6. , , 1 : 2, 3, 6, 7, 11.
:
| x1 x2 x3 x4 | f(x1 x2 x3 x4) | |
| 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 |
1 :
 (1)
 (1)
 (2)
 (2)
| 
| 
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:
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| |
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| + | ||||
|
| + | + | + | + |
f1(x1,x2,x3,x4) = 
˅
7. -, , 1 : 6, 7, 8, 10, 11, 13.
:
| x1 x2 x3 x4 | f(x1 x2 x3 x4) | |
| 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 |
1 :
1)
0 -
I - 1000
II 0110, 1010
III - 0111, 1011, 1101
IV -
2)
0 -
I - 10_0
II 011_, 101_
III -
:
| 10_0 | 011_ | 101_ | ||
| + | ||||
| + | ||||
| + | ||||
| + | ||||
| + | ||||
| + |
f1(x1,x2,x3,x4) = 
8. , , 1 : 2, 3, 4, 5, 10, 12, 13, 15.
:
| x1 x2 x3 x4 | f(x1 x2 x3 x4) | |
| 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 |
f1(x1,x2,x3,x4) = 
˅ 
˅ 
9. : , .
:
: V = {v1, v2, v3, v4, v5, v6,v7}
: = { x1, x2, x3, x4, x5, x6,x7, x8, x9}
A()7×7 = aij = 
:
B()6×95 = bij = 
| x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | |
| v1 | |||||||||
| v2 | -1 | -1 | |||||||
| v3 | -1 | -1 | |||||||
| v4 | |||||||||
| v5 | |||||||||
| v6 | -1 | ||||||||
| v7 | -1 | -1 | -1 | -1 |
10. :
;
;
;
;
.
:
0
9
7
1
| v1 | v2 | v3 | v4 | v5 | v6 | v7 | |
|
|
|
μ() :
μ() = |E()| - |V()| + k,
|E()| - .
|V()| - .
k .
, μ() = |E()| - |V()| + k = 9 7 + 1 = 3
11. , :
( );
( , );
( , , );
, ( ).
:
, .
.
, , - - , , .
,
, . .
.
, . .
. .
12. : , , , .
, , G, D(G) , dij vi vj. .
D(G) = 
G : r(vj) = max d(vi,vj) i, j = 1, 2, ,5. : r(v1) = 0, r(v2) =2, r(v3) =1, r(v4) = 0, r(v5) =0, r(v6) = 1, r(v7) = 1.
G, G. , R(G) = 0 D(G) = 2, v1, v4, v5.
13. ,
D = U(A∩B)
:
Program d;
uses crt;
const a=[1,2,3,5,7,8,10]
b=[1,2,3,5];
var r:set of byte;
i: byte;
begin
clrscr;
writeln;
r:=a-b;
for i:=1 to 255 do
var r:set of byte;
i: byte;
begin
clrscr;
writeln;
d:=a+r;
for i:=1 to 255 do
if i in r then write(i, );
readln;
end.
14. , : P Í = {(x, y): x + y = 0, x, y Î , }. , 10 .
, .
:
A = { -3; -2; -1; 1; 2; 3; 4; 5; 6; 7}
P = (-3;3), (-2;2), (-1;1),(1;-1),(2;-2),(3;-3)
:
1) 
-
2)
-
= 1 (-1;1) .
3)
-
= -1 y = 1 (-1;1)
4)
-
5)
-
6)
, .
|
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7)
-
= -1 y = 1 z = 7 (-1;1) , (1;7)
8)
-
= 1 y = 3 1≠3 (1;3) (3;1)
++
| #include <iostream> #include <conio.h> #include <stdio.h> #include <fstream> #include <locale.h> #include <math.h> using namespace std; // void Display(int A[],int k, int m[4][4]); void Display(int A[],int k, int** m); // void SortMas(int a[], long N){ long i = 0, k = N; int temp; bool flag = 0; while (flag!= true){ flag = true; for (i=0; i < k-1; i++){ if (a[i] > a[i+1]){ temp = a[i]; a[i] = a[i+1]; a[i+1] = temp; flag = false; } } } } // c void MakeSet(int m[], int k, int n){ int i, j; srand(time(0)); for (i = 0; i < k; i++){ m[i] = rand()%n+1; int j = 0; while (j < i){ if (m[i] == m[j]){ m[i] = rand()%n+1; j=-1; } j++; } } } //o void Display(int A[],int k, int** m){ //output of set cout<<"\n\n"; cout<<" |"; for (int i = 0; i < k; i++){ cout.width(3); cout<<A[i]<<" |"; } //print line cout<<"\n"; cout<<" "; for(int i = 0; i < k+1; i++){ cout<<"-----"; } cout<<"\n"; //print array for (int i = 0; i < k; i++){ cout<<" "; cout.width(4); cout<<A[i]<<" |"; for (int j = 0; j < k; j++){ cout.width(3); cout<<m[i][j]<<" |";; } if(i < k-1){ cout<<"\n"; cout<<" "; for(int i = 0; i < k+1; i++) { cout<<"-----"; } } cout<<"\n"; } //print line cout<<" "; for(int i = 0; i < k+1; i++){ cout<<"-----"; } } void Display(int A[],int k, int m[4][4]){ //output of set cout<<"\n\n"; cout<<" |"; for (int i = 0; i < k; i++){ cout.width(3); cout<<A[i]<<" |"; } //print line cout<<"\n"; cout<<" "; for(int i = 0; i < k+1; i++){ cout<<"-----"; } cout<<"\n"; //print array for (int i = 0; i < k; i++){ cout<<" "; cout.width(4); cout<<A[i]<<" |"; for (int j = 0; j < k; j++){ cout.width(3); cout<<m[i][j]<<" |";; } if(i < k-1){ cout<<"\n"; cout<<" "; for(int i = 0; i < k+1; i++) { cout<<"-----"; } } cout<<"\n"; } //print line cout<<" "; for(int i = 0; i < k+1; i++){ cout<<"-----"; } } // int ref (int m[4][4], int k){ int q; for (int i = 0; i<k; i++){ if (m[i][i]==0){ // 0 - q=0; break; } else q=1; // - } return q; } // int antiref (int m[4][4], int k){ int q; for (int i = 0; i<k; i++){ if (m[i][i]==1){ // 0 - q=0; break; } else q=1; // } return q; } // int sym (int m[4][4], int k){ int q=0; for (int i = 0; i<k; i++){ for (int j = 0; j<k; j++){ if (m[i][j]==m[j][i] && m[i][j]==1 && m[j][i]==1 && i!= j) q++; } } if (q>0) return 1; // 1 else return 0; } // int antisym (int m[4][4], int k){ int q; for (int i = 0; i<k; i++){ for (int j = 0; j<k; j++){ if ((m[i][j]==m[j][i]==1 && i==j) || (m[i][j]==0 && i!=j) || (m[i][j]!=m[j][i])) q=1; // , // else { q=0; break; } } if (q==0) break; } return q; } // // - int trans (int a[4][4], int k){ int mat[4][4]; int q=0; // mat[4][4] for (int i = 0; i<4; i++) for (int j = 0; j<4; j++) mat[i][j]=a[i][j]; // - . // mat[4][4] for (int l = 0; l<k; l++) for (int i = 0; i<k; i++) for (int j = 0; j<k; j++) mat[i][j] = (mat[i][j] || (mat[i][l] && mat[l][j])); // mat[4][4] [4][4] // , [4][4] , - . for (int i = 0; i<4; i++) for (int j = 0; j<4; j++){ if (mat[i][j] == a[i][j]) q=1; else { q=0; return q;} } return 1; } int main (){ srand(time(0)); int key; int i; int Na; do{ system("cls"); cout<<"\n"; cout<<" Please, input size set A --> "; cin>>Na; }while (Na<1 || Na>10); int *A = new int[Na];// int **M = new int *[Na]; // for (int i = 0; i<Na; i++){ M[i] = new int [Na]; } int M2[4][4] = { 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 }; int A2[4] = {1, 2, 3, 4}; MakeSet(A,Na,15);// SortMas(A,Na); // do { system("cls"); cout<<"\n\n\n"; cout<<" MENU: \n"; cout<<" 1) Output the matrix of truth for the relationship antisymmetry... \n"; cout<<" 2) Output the matrix of truth for the relationship transitivity... \n"; cout<<" 3) Check the properties of relations... \n"; cout<<" If you want to EXIT input 0\n"; cout<<" Please make your choice. --> "; cin>>key; // putchar (key); switch (key){ case 1:{ system("cls"); cout<<"\n DEMO of antisymmetry relation: R = ((b >= a) && (a+b)^2 % 3!= 0)? 1: 0"; for (int i = 0; i<Na; i++){ for (int j = 0; j<Na; j++){ M[i][j] = ((A[j] >= A[i]) && (A[i]+A[j])*(A[i]+A[j]) % 3!= 0)? 1:0; } } Display(A,Na,M); cout<<"\n\n Press any key!> "; getch(); break; } case 2:{ system("cls"); cout<<"\n."; cout<<"\n DEMO of transitivity relation: R = A:B? 1: 0"; for (int i = 0; i<Na; i++){ for (int j = 0; j<Na; j++){ M[i][j] = (A[i] % A[j] == 0)? 1:0; } } Display(A,Na,M); cout<<"\n\n Press any key!> "; getch(); break; } case 3:{ system("cls"); cout<<"\n Check the properties of relation R(A)={{1,1},{1,2},{2,1},{2,2},{3,3},{4,4}}:"; Display(A2,4,M2); cout<<"\n This relation is: "; if (ref(M2,4)) cout<<"\n -reflexive"; if (antiref(M2,4)) cout<<"\n -antireflexive"; if ((ref(M2,4) + antiref(M2,4))==0) cout<<"\n -not refleksive not antirefleksive"; if (sym(M2,4)) cout<<"\n -symmetric"; if (antisym(M2,4)) cout<<"\n -antisymmetric"; if (trans(M2,4)) cout<<"\n -transitive"; else cout<<"\n -not transitive"; cout<<"\n\n Press any key!> "; getch(); break; } } } while (key!= 0); delete []A; for (int i = 0; i<Na; i++){ delete[] M[i]; } delete[] M; return 0; } |
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