Equality
$$A \neq B$$
A is not equal to B
Subset
$$A \not\subseteq B$$
A is not a subset of or equal to B
Superset
$$A \not\supseteq B$$
A is not a superset of or equal to B
Intersection
$$A \cap B =$$
{ }
Union
$$A \cup B =$$
{ archetypical, earth, empirical, fire, metal, scientific, water, wood }
Symmetric Difference
$$A \ominus B =$$
{ archetypical, earth, empirical, fire, metal, scientific, water, wood }
Difference
$$A - B =$$
{ earth, fire, metal, water, wood }
Difference
$$B - A =$$
{ archetypical, empirical, scientific }
Cartesian Product
$$A \times B =$$
{ (earth, archetypical),
(earth, empirical),
(earth, scientific),
(fire, archetypical),
(fire, empirical),
(fire, scientific),
(metal, archetypical),
(metal, empirical),
(metal, scientific),
(water, archetypical),
(water, empirical),
(water, scientific),
(wood, archetypical),
(wood, empirical),
(wood, scientific) }