Formula Sheet by Abdullah Sheriff
Binary Operation: Must satisfy closure.
*Semi-Group (S, ): Must satisfy closure & associativity.
(Z, +) (Z, -) (R, x) (Pn(x), +)→ Semigroups
($Z^+$, -) ($P_n$(x), x)→ Not semigroups [No closure]
*Group (G, ): Must satisfy closure, associativity, existence of identity element & existence of inverse.
(Z, +) (R, +) (R - {0}, x) ($M_{2x2}$, +) ($M_{2x2}$ (Non-singular matrices), x)→ Groups
(R, x) (Z, x) (W, +) → Not groups [No inverse]
(N, +) → Not a group [No identity element]
*Abelian Group (G, ): Must satisfy closure, associativity, existence of identity element, existence of inverse & commutativity.
($M_{2x2}$, x) → Not an abelian group [∀ A, B ∈ $M_{2x2}$ ; A x B ≠ B x A]
*Ring (R, +, ): Must satisfy