Discrete Mathematics

A binary operation * on

A binary operation * on a non-empty set S is a rule that assigns to each ordered pair (a,b) ∈ S × S a unique element a*b ∈ S. This ensures closure pro

Every binary operation can satisfy various

Every binary operation can satisfy various properties: (1) Closure: a*b ∈ S for all a,b ∈ S; (2) Commutative: a*b = b*a; (3) Associative: a*(b*c) = (a

An algebraic structure is a non

An algebraic structure is a non-empty set on which one or more binary operations are defined with specific properties. Examples include: (ℤ, +) with c

A Boolean matrix contains only entries

A Boolean matrix contains only entries of 0 and 1. Two operations are defined on boolean matrices: Join (∨): c_ij = 1 if either a_ij = 1 or b_ij = 1 (

Modular arithmetic involves finding remainders when

Modular arithmetic involves finding remainders when dividing by a modulus n. We write a ≡ b (mod n) if a - b = nk for some integer k, meaning a and b