Vyberte vlastnosti, ktoré má uvedená relácia.

Na množine M = {1, 2, 3, 4, 5, 6} doplňte nasledujúcu reláciu tak, aby bola reflexívna. R={[1, 2], [2, 3], [3, 3], [3, 4], [2, 2], [6, 5], …}
- [1, 1],[4, 4],[5, 5],[6, 6]
- [2, 1], [3, 2],[4, 3],[5, 6]
- [1, 3], [2, 4]
- [1, 3], [1, 4],[1, 5],[1, 6],[2, 4],[2, 5], [2, 6],[3, 5],[3, 6], [4, 5], [4, 6],[5, 6]
Daná je množina M = {x, y, z, v, u}. Zistite, či relácia R={[x, x], [y, y], [z, z], [y, z], [z, u], [u, z], [z, y], [v, v], [u, u], [y, u], [u, y] } je reláciou ekvivalencie a vyberte príslušný rozklad množiny M.
Nech je daná množina A = {a, b, c}. Vyberte k rozkladu S = {{a, b}, {c}} príslušnú reláciu ekvivalencie.
- R ={[a, a],[b, b],[c, c],[a, b],[b, a] }
- R = {[a, a],[b, b],[c, c],[a, b],[b, a], [b, c], [c, b],[a, c], [c, a] }
- R = {[c, c],[a, b],[b, a] }
- R = {[c, c],[a, b],[b, a],[b, c], [c, b],[a, c], [c, a] }
Dané sú množiny A = {1, 2, 3, 4, 5} , B = {a, b, c, d}, C = {x, y, z}. Vyberte karteziánsky súčin C x A.
- {[x, 1], [x, 2], [x, 3],[x, 4], [x, 5], [y, 1], [y, 2], [y, 3],[y, 4], [y, 5][z, 1], [z, 2], [z, 3],[z, 4], [z, 5] }
- {[1, x], [2, x], [3, x],[4, x], [5, x], [1, y], [2, y], [3, y],[4, y], [5, y], [1, z], [2, z], [3, z],[4, z], [5, z] }
- {[x, x], [y, y], [z, z],[1, 1], [2, 2], [3, 3], [4, 4], [5, 5] }
- {[c, x], [b, x], [c, x],[d, x], [a, y], [b, y], [c, y],[d, y], [a, z], [b, z], [c, z],[d, z] }
Vyberte reláciu, ktorá je v množine M = {m, n, o, p} reflexívna a symetrická.
- {[m, n], [n, m],[p, m], [p, o], [o, m], [m, o], [o, p] , [m, p]}
- {[m, n], [n, m],[p, p], [p, o], [m, m], [o, o], [n, n] }
- {[m, n], [n, m], [o, p],[p, p], [p, o], [m, m], [o, o], [n, n] }
- {[m, n], [n, m],[p, p], [o, p],[p, o], [m, m], [o, o] }
Vyberte správnu definíciu. Reláciu R definovanú v množine M nazývame antisymetrickou, ak:
- ∀x,y∈R: [x,y]∈R=>[y,x]∈R
- ∀x,y∈M:x≠y∧ [x,y]∈R=>[y,x]∈R
- ∀x,y∈M:x≠y=> [x,y]∈R∨[y,x]∈R
- ∀x,y∈M:x≠y∧ [x,y]∈R=>[y,x]∉R