Chapter 2.3 — Exercise 2.2 — Types of Sets and Venn Diagrams

Empty Set and Universal Set

A set with no elements at all is called an empty set or null set, written φ or { }. A set of natural numbers strictly between 5 and 6 is empty — no such number exists. So is {x : x + 3 = 0 and x ∈ W}, since the only solution, −3, isn't a whole number. A universal set (written μ or U) is the opposite extreme: the complete collection of everything relevant to a given problem — if you're studying a school's students, the set of all students in that school is the universal set, and in a Venn diagram it's always the enclosing rectangle.

Two symbols are easy to confuse but mean very different things: φ, { }, and 0 all look related, but only φ and { } denote the empty set — 0 is a number, and {0} is actually a non-empty set containing the single element 0.

Subsets and How Many Exist

Set A is a subset of set B (written A ⊆ B) if every element of A also appears in B. Two rules apply universally: the empty set is a subset of every set, and every set is a subset of itself. The familiar number-system chain from earlier chapters is really one long subset relationship:

If a set has n elements, it has exactly 2ⁿ possible subsets, including the empty set and the full set itself:

Drawing Sets: The Venn Diagram

A Venn diagram (or Venn–Euler diagram) draws sets as circles inside a rectangle representing the universal set. Overlapping circles show shared elements; separated circles show none.

When two sets share no elements at all — called disjoint sets — their circles are drawn separately with no overlap, and A ∩ B = φ for that pair.

Union, Intersection, and Difference

Three operations combine two sets into a new one, each with its own symbol and meaning:

• Union (A ∪ B) — everything in A, or B, or both, with no element repeated. A ∪ B = {x : x ∈ A or x ∈ B}. If A ⊂ B, then A ∪ B = B, and A ∪ φ = A always.
• Intersection (A ∩ B) — only what's shared by both. A ∩ B = {x : x ∈ A and x ∈ B}. A ∩ φ = φ, A ∩ A = A, and if B ⊂ A then A ∩ B = B.
• Difference (A − B) — what's in A but not in B. A − B = {x : x ∈ A and x ∉ B}. Unlike the other two, difference is not commutative: A − B usually does not equal B − A.

Four Quick Checks

A handful of short examples confirm how these operations behave in practice:

• {1, 3, 7, 8} ∪ {2, 4, 7, 9} = {1, 2, 3, 4, 7, 8, 9} — every element from both sets, 7 written only once.
• {6, 9, 11} ∪ φ = {6, 9, 11} — unioning with the empty set changes nothing.
• {1,…,10} ∩ {2, 3, 5, 7} = {2, 3, 5, 7} — since the second set is entirely contained in the first, the intersection equals the smaller set.
• {4, 5, 6} ∪ {7, 8} = {7, 8} ∪ {4, 5, 6} = {4, 5, 6, 7, 8} — union doesn't care which set comes first.

Working Through the Exercise's Own Questions

With A = {1, 2, 3, 4} and B = {1, 2, 3, 5, 6}, both A ∩ B and B ∩ A equal {1, 2, 3} — confirming intersection is commutative, just like union. Taking A = {0, 2, 4}: A ∩ φ = φ and A ∩ A = A, matching the general rules above exactly. And for A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15}: A − B = {2, 4, 8, 10}, while B − A = {3, 9, 12, 15} — two different answers, the clearest possible demonstration that difference isn't commutative.

One more identity worth confirming directly: if A ⊂ B, taking A = {2, 4, 6} and B = {2, 4, 6, 8, 10} gives A ∪ B = {2, 4, 6, 8, 10} = B exactly — whenever one set already contains the other, their union is simply the larger set.

Six Intersections From Four Number Sets

With A = natural numbers, B = even naturals, C = odd naturals, and D = primes, every pairwise intersection reduces to a familiar set:

Ten Differences From Four Given Sets

With A = {3,6,9,12,15,18,21}, B = {4,8,12,16,20}, C = {2,4,6,8,10,12,14,16}, and D = {5,10,15,20}, the exercise asks for all ten possible differences between pairs:

B − C shrinking down to the single element {20} is worth a second look: every other element of B (4, 8, 12, 16) is also present in C, so only 20 survives the subtraction.

Disjoint or Not? Four Judgment Calls

The closing question asks whether four given pairs are disjoint — and the answer hinges entirely on whether they share even one element.

Continuing On

The subset idea introduced here — A ⊆ B meaning every element of A sits inside B — is exactly what Exercise 2.3 builds on next, defining two sets as equal precisely when each is a subset of the other. For the roster-and-set-builder groundwork this exercise assumes, see Exercise 2.1, or start from the chapter introduction.