Concepts: subsets, intervals, universal set and Venn diagrams
Sets24 min readLesson 3 of 7
3.1 Empty, finite and infinite sets
An empty set has no elements at all, and it is written with the Greek letter phi or as a pair of braces with nothing inside. The distinction that trips students is between a set that is hard to count and a set that is truly empty. The set of students studying in both Class X and Class XI is empty, because school rules do not allow the double enrolment. The set of natural numbers between 1 and 2 is empty, because there is no natural number strictly between them.
A set is finite when it is empty or has a definite number of elements, and infinite otherwise. The days of a week form a finite set, while the points on a line form an infinite set. When you write an infinite set in roster form, the three dots carry the pattern.
The counting symbol n(S) means the number of distinct elements of S. It is defined only for finite sets, so you cannot write n(N) as a natural number.
3.2 Subsets
A set A is a subset of B when every element of A also lies in B. In symbols, A inside B means that x in A implies x in B. The empty set is a subset of every set, and every set is a subset of itself.
When A is a subset of B and the two are not equal, A is a proper subset. A set with exactly one element is a singleton. The set of all divisors of 56 contains the set of prime divisors of 56, so the second is a subset of the first. The set of vowels is not a subset of {a, b, c, d} and neither is the reverse, because each set has an element missing from the other.
The most common trap in this section is the difference between belongs to and is a subset of. For a set A = {1, {3, 4}, 5}, the object {3, 4} is an element of A, so the statement {3, 4} in A is true, while {3, 4} subset of A is false because 3 and 4 are not elements of A. When a set contains a set as an element, read the braces before deciding.
3.3 Subsets of the real line and intervals
The real numbers contain a nested chain of important subsets. Natural numbers sit inside integers, integers sit inside rationals, and rationals sit inside reals. Irrational numbers form a separate branch inside the reals.
Intervals are the connected pieces of the real line. Four bounded types exist for real a < b: two closed at one or both ends, one open, and the half open pair. A round bracket leaves its end out and a square one takes it in. The length of each is b - a. Unbounded intervals run to infinity on one or both sides, and infinity always pairs with a round bracket.
One useful rule: if two intervals are nested, the smaller one is a subset of the larger. So (-3, 5) lies inside [-7, 9] because every point strictly between -3 and 5 also lies between -7 and 9. Checking endpoints is enough for nested intervals, but for intervals that stick out at one end, compare the ranges.
3.4 The universal set
Every discussion of sets happens inside a larger frame. In a problem about divisors of 42, the frame might be the set of natural numbers. In a survey of Class XI students, the frame is those students. This frame is the universal set, written U, and all sets in the problem are subsets of it.
The universal set is chosen by context, not by a fixed rule. For the set of integers, the frame can be the rationals or the reals. For letters of a word, the frame is the English alphabet. When a question asks you to propose a universal set, choose one that contains every set mentioned in the problem, then check that no element of the universal set is missing from the union of the given sets.
The complement operation, defined later in the chapter, has no meaning until U is fixed. That is why exam questions usually state the universal set first.
3.5 Venn diagrams
A Venn diagram draws the universal set as a rectangle and each subset as a closed curve, usually a circle. Elements are written inside the region where they belong. The diagram is named after the English logician John Venn.
The picture earns its place when a question asks you to show a relationship rather than compute one. A circle drawn inside another shows a subset relation. Two circles that do not touch show disjoint sets. Two overlapping circles split the rectangle into four regions: only A, only B, both, and neither.
The diagram is a sketch of the relationship, not a proof. To prove a set identity you still need element chasing or the set laws; the diagram only tells you which identity to try.
Exam
When a question says write all subsets, count first. A finite set with n elements has exactly 2 to the power n subsets, so {a, b} has four: phi, {a}, {b} and {a, b}. If your list has a different count, something is missing or repeated.
3.6 Worked checks for this page
Exercise 1.2 tests empty, finite and infinite sets, and Exercise 1.3 tests subsets, intervals and universal sets. Try these three shapes.
First, is the collection of even primes above 2 empty? Yes, because no even number larger than 2 is prime, and the lone even prime, 2, is ruled out by the condition. Second, list every subset of the three element set. The count is eight: the empty one, three singletons, three pairs and the full set. Third, turn the real numbers satisfying -4 < x <= 6 into interval notation. The left end is left out and the right end is taken in, so the interval is (-4, 6].
In each case the method is the same: read the condition, decide membership at the boundary, then write the answer in the form the question asked for.