Reflexive, Symmetric, Transitive, Associate, Commutative, Distributive Laws and Operators

A refresher:

Reflexive: a = a
(Every element is mapped back to itself)

Symmetric:  a = b,  => b = a

Transitive: a = b  & b = c, => a = c

Associative: (x # y) # z = x # (y # z)
(The operator is immune to different grouping of the operands)

Commutative: x $ y = y $ x
(The operator is immune to changing order of operands)

Distributive: x @ (y ^ z) = x @ y ^  x @ z,
(Outer Operand x can be distributed over the other two y & z without affecting the results)