SET-THEORETIC NOTATION - RECENT DEVELOPMENTS - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER IX. RECENT DEVELOPMENTS

§5. SET-THEORETIC NOTATION

Mathematical notation follows fashions, and sometimes the fashion can change. In consequence, Courant and Robbins’s terminology occasionally differs in very minor ways from what is now current, but this is seldom important enough to mention (e.g., “Hypothesis of the Continuum” instead of “Continuum Hypothesis”). On this particular occasion, however, the difference from current practice is too significant to be ignored.

The terms “logical sum” and “logical product” are hardly ever used nowadays; instead, the alternatives “union” and “intersection” are employed. The empty set is denoted Ø, not O, and there is no longer a special symbol I for the universe of discourse. The current notations for the union and intersection of two sets A and B are as follows:

Union: AB (in place of Courant and Robbins’s A + B)

Intersection: AB (in place of Courant and Robbins’s AB).

The complement A’ is often written Ac, but A′ is still common. The current notation for subsets is either ⊂ or ⊆. Unlike < and ≤, the expression AB does not imply that AB, either today or in Courant and Robbins’s time. In order to denote inequality in a subset relation, the cumbersome notation A ⊆ B is used.

The notations A + B, AB, and A’ do still survive in computer science and electronic engineering, where they are used to describe circuits formed from logic gates.

Ironically, the modern notation obscures the algebraic analogies in properties (6–17) on p. 110. In view of (10, 11, 13), however, this may not be entirely a bad thing.