Who's In, Who's Out

A set is a well-defined collection of distinct objects: given any object, it must be possible to say for certain whether it belongs to the collection or not. 'The set of all factors of 24' is well-defined, since any number can be checked for whether it divides 24 exactly. 'The set of the most interesting numbers' is NOT well-defined, since 'interesting' means different things to different people, and there is no fixed test to decide membership. A set can be written in roster form, listing every element inside braces, such as the factors of 24 written as {1, 2, 3, 4, 6, 8, 12, 24}, or in set-builder form, describing a shared property instead of listing every member, such as {x : x is a natural number and x divides 24}. Both notations describe the exact same set; roster form is convenient for small, listable sets, while set-builder form is the only real option once a set has infinitely many members.

Consider the set {x : x is a natural number and 5 < x < 6}. No natural number lies strictly between 5 and 6, so this set has no elements at all: it is called the empty set, written as either the symbol Ø or as empty braces { }. Contrast this with the set of days of the week, {Monday, Tuesday, ..., Sunday}, which has exactly 7 elements: a finite set. The set of even natural numbers, {2, 4, 6, 8, ...}, never stops: it is an infinite set, and cannot ever be fully written out in roster form, only suggested with a pattern and three dots, or captured completely and unambiguously in set-builder form as {x : x is a natural number and x is even}.

A set A is a subset of a set B if every single element of A is also an element of B, written A subset-of B. The set of all squares is a subset of the set of all rectangles, since every square genuinely is a rectangle (four right angles, opposite sides equal), but the reverse is false: a rectangle with unequal adjacent sides is not a square, so the set of rectangles is NOT a subset of the set of squares. A particularly important family of subsets of the real numbers R are intervals. A safe vaccine-fridge temperature range, say from 2 degrees C to 8 degrees C inclusive, is the closed interval [2, 8] = {x : 2 <= x <= 8}, including both endpoints. A range that excludes an endpoint, such as 'strictly less than 8 degrees C but at least 2 degrees C,' is the half-open interval [2, 8) = {x : 2 <= x < 8}. Every interval is really just a subset of R described by an inequality.

Every discussion of sets happens inside some larger, agreed-upon universal set U, the set of everything under consideration in that context. In a school survey, U might be every student in the school, with smaller sets like the students who play cricket living inside it as subsets. A Venn diagram, named after the English logician John Venn, pictures this: U is drawn as a rectangle, and its subsets as circles inside it. Two overlapping circles inside the same rectangle can show two subsets of U at once, with the overlapping region representing whatever belongs to both. This simple picture makes the next few operations, ways of building new sets out of old ones, easy to see rather than just easy to state.

In a survey of a class, let M be the set of students who like Maths and S be the set of students who like Science. The union, M union S, is the set of students who like Maths OR Science (including those who like both): every student in M, every student in S, with anyone appearing in both counted only once. The intersection, M intersect S, is the set of students who like BOTH Maths AND Science: only the overlap. If M = {Aisha, Rohan, Priya, Karan} and S = {Rohan, Priya, Meera}, then M union S = {Aisha, Rohan, Priya, Karan, Meera} and M intersect S = {Rohan, Priya}. Union and intersection both obey a commutative law (M union S = S union M, and likewise for intersection) and an associative law, and intersection distributes over union: M intersect (S union T) = (M intersect S) union (M intersect T) for any third set T, a law that can be checked directly on a Venn diagram by shading both sides and seeing they match.

The difference M minus S is the set of students who like Maths but NOT Science: removing anyone who appears in S from M. Using the same class, M minus S = {Aisha, Karan} (Rohan and Priya are removed since they also like Science), while S minus M = {Meera} (only Meera remains once Rohan and Priya, who also like Maths, are removed). Notice M minus S is generally not equal to S minus M: difference depends on order, unlike union or intersection. The complement of a set A, written A', is everything in the universal set U that is NOT in A: A' = U minus A. If U is the whole class of 10 students and M (Maths-likers) has 4 of them, then M', the 6 students who do NOT like Maths, depends entirely on which U is chosen: change the universal set, and the complement changes too, even though M itself hasn't moved.

What does it mean to be outside M union S? Exactly this: NOT liking Maths and NOT liking Science, at the same time, since if a student liked even one of the two subjects they would already be inside the union. This everyday reasoning is exactly De Morgan's first law: (M union S)' = M' intersect S'. Checking directly with U as the full class of 10, if M union S = {Aisha, Rohan, Priya, Karan, Meera} (5 students), then (M union S)' is the other 5 students, those liking neither subject. Separately, M' is everyone except {Aisha, Rohan, Priya, Karan}, and S' is everyone except {Rohan, Priya, Meera}; intersecting M' and S' (people missing from both lists) gives exactly those same 5 students. The second law works the same way in reverse: (M intersect S)' = M' union S', being outside the overlap means missing from at least one of the two sets. These laws are named after the mathematician De Morgan, though set theory itself traces back to Georg Cantor's work in the 1870s, later surviving the famous Russell's Paradox that shook its foundations around 1902, before being rebuilt on firmer axioms in the early 20th century.