What is a Set? — Definition and Notation

A set is a well-defined collection of distinct objects. "Well-defined" means that, given any object, we can say with certainty whether it belongs to the collection or not — "the set of vowels in the English alphabet" is well-defined (we can check any letter against it), but "the set of intelligent students in a class" is not, because "intelligent" has no fixed test. This distinction is what separates a mathematical set from a loose everyday collection.

Sets are usually named with a capital letter — $A$, $B$, $X$ — and written by enclosing their objects (called elements or members) in curly braces $\{\ \}$. If $a$ is an element of a set $A$, we write $a \in A$ (read "$a$ belongs to $A$" or "$a$ is a member of $A$"); if $a$ is not an element of $A$, we write $a \notin A$.

For example, if $A = \{2, 3, 5, 7\}$ (the set of prime numbers less than 10), then $3 \in A$ but $4 \notin A$. Two sets are considered equal if they contain exactly the same elements, regardless of the order in which those elements are listed — $\{1, 2, 3\}$ and $\{3, 1, 2\}$ are the same set.

This chapter builds the vocabulary — sets, their representation, special sets, cardinality, Cartesian products, and subset relations — that every later topic in this WBCHSE Class 11 Business Mathematics and Basic Statistics syllabus (probability, functions, set operations) is built on top of. Getting comfortable with these building blocks now makes the West Bengal HS commerce mathematics syllabus's later, more applied set-theory content far easier to follow.