Sets

A set is a well-defined collection of distinct objects. "Well-defined" means that, given any object, you can decide without ambiguity whether or not it belongs to the collection. The objects in a set are its elements (or members). We write $a \in A$ for "$a$ is an element of $A$" and $a \notin A$ for "$a$ is not an element of $A$".

"The collection of tall students" is not a set — "tall" is subjective. "The collection of students above 170 cm" is a set. Sets are named with capital letters; elements are usually written in lower case.

Two ways to represent a set:

In roster (tabular) form you list every element inside braces, separated by commas. Order does not matter and elements are never repeated — $\{1,2,2,3\}$ is just $\{1,2,3\}$. In set-builder form you state the property all elements share, read as "the set of all $x$ such that …".

Standard number sets have reserved symbols you will use throughout the syllabus:

Types of sets by size: The empty set $\varnothing$ (or $\{\,\}$) has no elements. A singleton has exactly one. A set is finite if its elements can be counted and the counting ends, and infinite otherwise. The number of elements in a finite set $A$ is its cardinal number, written $n(A)$.

Equal sets: $A = B$ exactly when they have precisely the same elements — every element of $A$ is in $B$ and vice versa. Equivalent sets only need the same cardinal number: $\{a,b,c\}$ and $\{1,2,3\}$ are equivalent but not equal.

Deeper Insight — why "well-defined" and "distinct" carry the whole subject: Every idea in this chapter rests on the two words in the definition. Well-defined guarantees that membership is a yes/no question with no grey area, which is exactly what lets us reason about sets with certainty — and it is why a vague collection like "good books" can never be a set. Distinct is the reason $\{1,1,2\}=\{1,2\}$: a set records only which objects are present, never how many times or in what order. Confusing a set with a list (where order and repetition matter) is the single most common beginner error, and it quietly breaks later work — for instance, counting in Permutations and Combinations depends on knowing exactly when arrangement matters and when only membership does. Treat a set as a pure "yes/no membership table" and the operations that follow become almost obvious.

$A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$. Formally:

If $A \subseteq B$ but $A \ne B$ (so $B$ has at least one element not in $A$), then $A$ is a proper subset, written $A \subset B$. Two facts hold for every set $A$: the empty set is a subset of it ($\varnothing \subseteq A$), and every set is a subset of itself ($A \subseteq A$).

Universal set $U$ is the fixed "background" set containing all objects under discussion for a particular problem. Every other set in that problem is a subset of $U$.

Intervals are subsets of $\mathbb{R}$ describing continuous stretches of the number line. A square bracket includes the endpoint; a round bracket excludes it:

Power set: the set of all subsets of $A$, written $P(A)$. Its elements are themselves sets. The count of subsets is fixed by the size of $A$:

The reason is a counting argument: to build a subset you make one independent yes/no choice per element — "include it or not" — giving $2 \times 2 \times \dots \times 2 = 2^m$ possibilities. Of these, $2^m - 1$ are proper subsets (all except $A$ itself).

Deeper Insight — subset is the backbone, $2^m$ is its first dividend: The subset relation $\subseteq$ is the most important relation in set theory because nearly every later definition is phrased through it — equality ($A=B$ means $A\subseteq B$ and $B\subseteq A$), the complement, even the way a universal set frames a problem. Keep two edge cases permanently in mind: $\varnothing$ is a subset of everything (vacuously — there is no element of $\varnothing$ that could fail the test), and every set is a subset of itself. The power-set formula $2^m$ is your first real glimpse of why these ideas matter beyond bookkeeping: it is the same "independent binary choice" logic that powers the Binomial Theorem ($\sum \binom{m}{k} = 2^m$, the total number of subsets counted by size) and probability over finite outcomes. Once you see a subset as a string of yes/no decisions, $2^m$ stops being a formula to memorise and becomes something you can re-derive on the spot.

Four operations combine sets, all visualised cleanly with Venn diagrams (a rectangle for the universal set $U$, overlapping circles for the sets).

Union $A \cup B$ collects everything in either set (the inclusive "or"). Intersection $A \cap B$ keeps only what is common to both. When $A \cap B = \varnothing$ the sets share nothing and are called disjoint.

Difference $A - B$ keeps the part of $A$ outside $B$ (note $A - B \ne B - A$ in general). Complement $A'$ is everything in the universal set that is not in $A$.

These operations obey algebraic laws that mirror ordinary arithmetic:

Counting with the inclusion–exclusion formula: the size of a union is not simply $n(A) + n(B)$, because elements in both are counted twice. Subtract the overlap once:

For three sets the same logic gives $n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A\cap B) - n(B\cap C) - n(A\cap C) + n(A\cap B\cap C)$.

Deeper Insight — De Morgan and inclusion–exclusion are the two ideas worth real understanding: Most set-operation questions reduce to one of two things. First, De Morgan's laws capture how "not" interacts with "and"/"or": the complement of a union is the intersection of complements, and vice versa — the operation flips when negation passes through. This is not a quirk of sets; it is the same logic that governs negating compound statements ("not (rich and famous)" $=$ "not rich or not famous") and is reused directly in Probability and in digital logic. Second, the inclusion–exclusion formula exists for one reason: naive addition double-counts the overlap. Picture the Venn diagram and the correction term $-\,n(A\cap B)$ becomes self-evident — you are simply refusing to pay for the shared region twice. Almost every "how many students study at least one of …" word problem is this single formula in disguise, so understanding why the overlap is subtracted, rather than memorising the symbols, is what lets you extend it confidently to three sets and beyond.