Fundamental Laws of Set Algebra- Root Digging

This post provides an intuition about the names of the fundamental laws of Set Algebra.

A set can be viewed as any well-defined collection of objects. The objects in a set may or may not have similar properties. The word “set” has a Latin origin and is derived from the word “secta” or “sect”. The Latin root sect means “cut”. Many English words, such as section, sector, insect, dissect, and intersect also have this Latin root. The word sect also refers to a subgroup of a religious, political, or philosophical belief system.

Algebra of Sets

Sets under the operations of union, intersection, and complement satisfy various laws with well-established names. The roots of these names they borrow from other languages provide additional intuition and understanding of the underlying concepts.

Laws of the Algebra of Sets

Idempotent Laws
$$A\cup A =A \quad\quad\quad\quad A\cap A =A$$

Under Idempotent Law, the operations can be applied without changing the initial values of the operands. The term “Idem” has a Latin origin which means “the same”. The second part “potent” also has a Latin origin and comes from the Latin word “posse” which means “being powerful” or “being able”. Idempotent Laws show their power of remaining on the same values, irrespective of the operations being performed on it.

Associative Laws
$$(A\cup B)\cup C =A\cup (B\cup C) \quad\quad\quad\quad (A\cap B)\cap C =A\cap (B\cap C)$$

Under Associative law, the operands can be associated by operators in any way desired. The Latin root is “associare” which means “join with” or “to unite”. Under associative law the grouping (or association) of the operands does not affect the result.

Commutative Laws
$$A\cup B =B\cup A\quad\quad\quad\quad A\cap B = B\cap A$$

According to commutative laws, the order in which the operators are evaluated does not affect the result. The word “commute” comes from Latin word “commutare”", meaning “to change altogether”" (from com - ‘altogether’ + mutare ‘to change’). Therefore, under commutative property we can move or (change) operands around and still get the same results. The Latin root “idem” and then in Late Latin “identitas” means “same”.

Distributive Laws
$$A\cup( B \cap C) = (A \cup B) \cap (A\cup C)\quad\quad\quad\quad A\cap( B \cup C) = (A \cap B) \cup (A\cap C)$$

The word “Distributive” has a Latin origin and comes from the word “distribut”, meaning “divided up”. Under Distributive Law we “spread out” or “share” the initial part with all the other terms.

Identity Laws
$$A \cup \phi =A\quad\quad\quad\quad A \cap \phi =\phi$$

$$A \cup U =U\quad\quad\quad\quad A \cap U =A$$

$\phi$ and $U$ are the identity elements in set algebra. Under identity law the characteristics of the operands determine the owner or the holder of the final result. The word identity means “quality of being identical” and has a Latin origin.

Involution Law
$$(A^\prime)^\prime = A$$

The involution law states that the double complement of a set gives the same set. The word “involution” is derived from the Latin word “volvere”, meaning “to roll” (from in - ‘into’ + volvere - ‘to roll’). What happen if you roll up your t-shirt inside out twice.

$Figure: (a) $A$ , (b) $A^\prime$, (c) ($A^\prime$)^\prime$

Complement Laws

$$A \cup A^\prime =U\quad\quad\quad\quad \phi^\prime =U$$

The word “complement” is derived form the Latin root, “complere” which means “fill up” (from com - ‘expressing intensive force’ + plere - ‘fill’). As the name suggested, the above two expressions give the universal set, which contains all the objects under consideration.

The below expressions also in regards to the universal set which is the complete set of objects under consideration.

$$A \cap A^\prime =\phi \quad\quad\quad\quad U^\prime =\phi $$

DeMorgan’s Laws
$$(A \cup B)^\prime =A^\prime \cap B^\prime\quad\quad\quad\quad (A \cap B)^\prime= A^\prime \cup B^\prime$$

According to the De Morgan’s Law, the complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements. The law is named after the British mathematician and logician, Augustus De Morgan (1806 – 1871).

Reference:

Spiegel, M. R., Schiller, J. J., & Srinivasan, R. A. (2013). Schaum’s outline of probability and statistics. McGraw-Hill Education.