Venn Diagrams

Sets can be represented graphically using Venn diagrams, named after the English mathemati-cian John Venn, who introduced their use in 1881. In Venn diagrams the universal set U, which contains all the objects under consideration, is represented by a rectangle. (Note that the univer-sal set varies depending on which objects are of interest.) Inside this rectangle, circles or other geometrical figures are used to represent sets. Sometimes points are used to represent the par-ticular elements of the set. Venn diagrams are often used to indicate the relationships between sets. We show how a Venn diagram can be used in Example 7.

EXAMPLE 7 Draw a Venn diagram that represents V, the set of vowels in the English alphabet.

Solution:We draw a rectangle to indicate the universal set U, which is the set of the 26 letters of the English alphabet. Inside this rectangle we draw a circle to represent V. Inside this circle

we indicate the elements of V with points (see Figure 1).

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U

V a

e u

o i

FIGURE 1 Venn diagram for the set of vowels.

2.1.3 Subsets

It is common to encounter situations where the elements of one set are also the elements of a second set. We now introduce some terminology and notation to express such relationships between sets.

Definition 3

We see that A⊆ B if and only if the quantification

∀x(x ∈ A→ x ∈ B)

is true. Note that to show that A is not a subset of B we need only find one element x ∈ A with x ∉ B. Such an x is a counterexample to the claim that x ∈ A implies x ∈ B.

We have these useful rules for determining whether one set is a subset of another:

Showing that A is a Subset of B To show that A⊆ B, show that if x belongs to A then x also belongs to B.

Showing that A is Not a Subset of B To show that A⊈ B, find a single x ∈ A such that x ∉ B.

EXAMPLE 8 The set of all odd positive integers less than 10 is a subset of the set of all positive integers less than 10, the set of rational numbers is a subset of the set of real numbers, the set of all computer

Source: Library of Congress Prints and Photographs Division [LC-USZ62-49535]

BERTRAND RUSSELL (1872–1970) Bertrand Russell was born into a prominent English family active in Links

the progressive movement and having a strong commitment to liberty. He became an orphan at an early age and was placed in the care of his father’s parents, who had him educated at home. He entered Trinity College, Cambridge, in 1890, where he excelled in mathematics and in moral science. He won a fellowship on the basis of his work on the foundations of geometry. In 1910 Trinity College appointed him to a lectureship in logic and the philosophy of mathematics.

Russell fought for progressive causes throughout his life. He held strong pacifist views, and his protests against World War I led to dismissal from his position at Trinity College. He was imprisoned for 6 months in 1918 because of an article he wrote that was branded as seditious. Russell fought for women’s suffrage in Great Britain. In 1961, at the age of 89, he was imprisoned for the second time for his protests advocating nuclear disarmament.

Russell’s greatest work was in his development of principles that could be used as a foundation for all of mathematics. His most famous work is Principia Mathematica, written with Alfred North Whitehead, which attempts to deduce all of mathematics using a set of primitive axioms. He wrote many books on philosophy, physics, and his political ideas. Russell won the Nobel Prize for Literature in 1950.

science majors at your school is a subset of the set of all students at your school, and the set of all people in China is a subset of the set of all people in China (that is, it is a subset of itself).

Each of these facts follows immediately by noting that an element that belongs to the first set in each pair of sets also belongs to the second set in that pair.

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EXAMPLE 9 The set of integers with squares less than 100 is not a subset of the set of nonnegative integers because −1 is in the former set [as (−1)² < 100], but not the latter set. The set of people who have taken discrete mathematics at your school is not a subset of the set of all computer science majors at your school if there is at least one student who has taken discrete mathematics who is

not a computer science major.

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Theorem 1 shows that every nonempty set S is guaranteed to have at least two subsets, the empty set and the set S itself, that is, ∅⊆ S and S ⊆ S.

THEOREM 1 For every set S, (i ) ∅⊆ S and (ii ) S ⊆ S.

Proof:We will prove (i ) and leave the proof of (ii ) as an exercise.

Let S be a set. To show that ∅⊆ S, we must show that ∀x(x ∈ ∅ → x ∈ S) is true. Because the empty set contains no elements, it follows that x ∈ ∅ is always false. It follows that the conditional statement x ∈ ∅→ x ∈ S is always true, because its hypothesis is always false and a conditional statement with a false hypothesis is true. Therefore, ∀x(x ∈ ∅→ x ∈ S) is true. This completes the proof of (i). Note that this is an example of a vacuous proof.

When we wish to emphasize that a set A is a subset of a set B but that A≠ B, we write A ⊂ B and say that A is a proper subset of B. For A ⊂ Bto be true, it must be the case that A ⊆ B and there must exist an element x of B that is not an element of A. That is, A is a proper subset of B if and only if

∀x(x ∈ A→ x ∈ B) ∧ ∃x(x ∈ B ∧ x ∉ A)

is true. Venn diagrams can be used to illustrate that a set A is a subset of a set B. We draw the universal set U as a rectangle. Within this rectangle we draw a circle for B. Because A is a subset of B, we draw the circle for A within the circle for B. This relationship is shown in Figure 2.

Recall from Definition 2 that sets are equal if they have the same elements. A useful way to show that two sets have the same elements is to show that each set is a subset of the other.

In other words, we can show that if A and B are sets with A⊆ B and B ⊆ A, then A = B. That is, A = B if and only if ∀x(x ∈ A→ x ∈ B) and ∀x(x ∈ B → x ∈ A) or equivalently if and only if ∀x(x ∈ A↔ x ∈ B), which is what it means for the A and B to be equal. Because this method of showing two sets are equal is so useful, we highlight it here.

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JOHN VENN (1834–1923) John Venn was born into a London suburban family noted for its philanthropy.

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He attended London schools and got his mathematics degree from Caius College, Cambridge, in 1857. He was elected a fellow of this college and held his fellowship there until his death. He took holy orders in 1859 and, after a brief stint of religious work, returned to Cambridge, where he developed programs in the moral sciences.

Besides his mathematical work, Venn had an interest in history and wrote extensively about his college and family.

Venn’s book Symbolic Logic clarifies ideas originally presented by Boole. In this book, Venn presents a systematic development of a method that uses geometric figures, known now as Venn diagrams. Today these diagrams are primarily used to analyze logical arguments and to illustrate relationships between sets. In addition to his work on symbolic logic, Venn made contributions to probability theory described in his widely used textbook on that subject.

U

A B

FIGURE 2 Venn diagram showing that A is a subset of B.

Showing Two Sets are Equal To show that two sets A and B are equal, show that A⊆ B and B⊆ A.

Sets may have other sets as members. For instance, we have the sets

A = {∅, {a}, {b}, {a, b}} and B = {x ∣ x is a subset of the set {a, b}}. Note that these two sets are equal, that is, A = B. Also note that {a} ∈ A, but a ∉ A.