Venn Diagrams and Set Theory Vocabulary

Universal Set and Basic Notation

In this transcript, the instructor introduces Venn diagrams that compare two data sets. A Venn diagram uses a universal set, usually drawn as a rectangle, containing all elements under consideration, with two overlapping circles representing sets A and B. Elements inside circle A belong to set A, elements inside circle B belong to set B, and the overlapping region represents the intersection A ∩ B, which contains elements that are in both sets. The universal set is the collection of all elements being considered for the problem; in the first example, the universal set is the natural numbers less than 20, so it comprises the numbers 1 through 19 (since the problem states “less than 20”). The diagram helps visualize which elements belong to A, which belong to B, and which belong to both or neither.

Sets in a Venn Diagram: A, B, and the Universe

Two explicit sets are defined in the first example. Set A consists of the natural numbers less than 20 that are divisible by three. That gives A = {3, 6, 9, 12, 15, 18}. Set B consists of the natural numbers less than 20 that are divisible by five, so B = {5, 10, 15}. The universal set U is all natural numbers less than 20, i.e., U = {1, 2, …, 19}. The intersection A ∩ B is the set of numbers that are in both A and B, which in this case is {15}. Visualizing these in the Venn diagram shows A as one circle containing {3, 6, 9, 12, 15, 18}, B as another circle containing {5, 10, 15}, and their overlap containing the single element 15. The instructor emphasizes that Venn diagrams are tools for visualization and organization of sets, and the placement of elements depends on which sets are being considered.

Intersection, Union, and Complement

The transcript reinforces the standard set operations:

Intersection: A ∩ B contains elements that are in both A and B. In our example, A ∩ B = {15}.
Union: A ∪ B contains all elements that are in A or B (or both). In the example, A ∪ B = {3, 5, 6, 9, 10, 12, 15, 18}.
Complement: The complement A^c (with respect to the universal set U) contains all elements in U that are not in A. Similarly, B^c contains all elements in U not in B. For a problem where U = {1, 2, …, 19}, A^c would be U \ A and B^c would be U \ B.

The instructor also introduces combinations like A^c ∪ B (the complement of A together with B), which includes all elements not in A plus all elements in B. He notes that some expressions are easier to describe verbally or spell out in a diagram than to type on a computer, such as the intersection symbol, which students might handwritten or represented in multiple-choice form on tests.

Shading Regions on Venn Diagrams

A key skill is shading the region that represents a specified set. Examples mentioned include:

Shade the region that represents A (everything inside circle A).
Shade the region that represents B (everything inside circle B).
Shade A ∪ B (the entire area inside either circle A or circle B, including the overlap).
Shade A ∩ B (the overlapping region).
Shade A^c (everything outside circle A).
Shade B^c (everything outside circle B).
Shade (A^c) ∪ B (everything outside A or inside B).
Shade (A^c) ∩ B (the portion of B that is not in A).
Shade (A^c) ∪ (B^c) (the region outside A or outside B, which is the complement of A ∩ B).
Shade (A^c) ∩ (B^c) (the region outside both A and B). This corresponds to the complement of A ∪ B.
The instructor also demonstrates how to interpret these shaded regions in relation to the universal set and how certain regions relate to logical statements like De Morgan’s laws (e.g., the complement of an intersection is the union of the complements).

Worked Example 1: Natural Numbers Under 20 Divisible by 3 and 5

Set A is the natural numbers less than 20 that are divisible by 3: A = {3, 6, 9, 12, 15, 18}.
Set B is the natural numbers less than 20 that are divisible by 5: B = {5, 10, 15}.
The universal set is the natural numbers less than 20: U = {1, 2, …, 19}.
The intersection A ∩ B is the number(s) in both sets: A ∩ B = {15}.
Shading these regions on the Venn diagram would show A as the left circle containing 3, 6, 9, 12, 15, 18; B as the right circle containing 5, 10, 15; and the overlap containing 15. The union A ∪ B would shade all elements in either A or B (i.e., 3, 5, 6, 9, 10, 12, 15, 18). The complement regions would shade everything outside A and B within U, including numbers like 1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, depending on the exact shading target (A^c, B^c, etc.).

Worked Example 2: Shading Tasks and Nomenclature

The instructor uses a repeated Venn diagram and asks students to shade various regions corresponding to different set expressions. For example:

Shade the region representing A (the entire interior of circle A).
Shade the region representing B (the interior of circle B).
Shade the region representing A ∪ B (the union, i.e., everything in either circle).
Shade the region representing A ∩ B (the overlapping middle).
For the complement operations, shade everything outside a circle (A^c or B^c).
Shade (A^c) ∪ B, which includes all elements not in A plus all elements in B.
Shade (A^c) ∩ B, which is the portion of B that is not in A.
Shade (A^c) ∪ (B^c) (the region outside both circles, i.e., the complement of A ∪ B).
Shade (A^c) ∩ (B^c) (the region outside both A and B, equivalent to (A ∪ B)^c).
The instructor notes that cases like A^c ∩ B^c may be visualized as the area of the universal set not covered by either circle. He also points out practical note-taking tips, such as handwriting the intersection symbol on tests when typing is cumbersome.

Worked Example 3: A and B within a Universe U

Another example defines U as the natural numbers less than 10 and sets A and B as specific subsets of U. For instance, A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7, 9}, with U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. The Venn diagram would show A and B overlapping in {1, 3, 5}. The elements in A ∩ B are {1, 3, 5}. The elements in A ∪ B are {1, 2, 3, 4, 5, 6, 7, 9}. The complement A^c within U would be {7, 8, 9} if U is the nine-element universe. The instructor also asks to identify I = A ∩ B = {1, 3, 5}, A ∪ B = {1, 2, 3, 4, 5, 6, 7, 9}, and A^c = U \ A, noting that a misstatement might occur in live dialogue (the instructor initially suggested A^c = {8} for a specific subset, which would be inconsistent with U = {1,..,9}); the correct A^c in that setup is {7, 8, 9}.

Generalized Venn Diagrams: Subsets and Disjoint Sets

The transcript moves to generalized Venn diagrams without populated elements, focusing on structure. Two main ideas are covered:

If A ⊆ B (A is a subset of B), then in the Venn diagram A is entirely contained within B. The universal set U contains both A and B, with A drawn as a circle entirely inside the circle for B.
If B ⊆ A, then B is entirely contained within A, with A drawn as the outer circle containing B.
The instructor also discusses the case where A and B are disjoint (A ∩ B = ∅), meaning the two circles do not overlap.
A student asks whether there are multiple valid ways to draw a Venn diagram that express subset relationships, noting that sometimes the outer region can be labeled as the universal set. The instructor cautions that in standard notation, the outermost shape should be the universal set and inner shapes should reflect subset relations, keeping A inside B (or B inside A) depending on the subset relation described. He also explores a scenario where A ⊆ B is represented by a single circle A inside a larger circle B, and conversely B ⊆ A by placing B inside A. The conversation clarifies that the subset notation communicates a containment relationship rather than a count of elements; multiple diagram conventions can convey the same relationships, but consistency with the universal set is important.

Worked Example 4: Piano and Voice Lessons

In a word problem about a music teacher with 21 students, the teacher gives 16 piano lessons and 12 voice lessons, with 16 students taking piano and 12 taking voice, and the question asks how many students take both types of lessons. Let x denote the number of students who take both piano and voice lessons, i.e., the overlap A ∩ B. Then the number taking piano only is 16 − x, and the number taking voice only is 12 − x. Since every student takes at least one type of lesson, the total must satisfy
21 = (16 - x) + (12 - x) + x.
This simplifies to 21 = 28 − x, giving x = 7. Therefore, seven students take both piano and voice lessons. The instructor emphasizes best practices for solving such problems on tests: first set up the overlapping regions in the Venn diagram, then use the total to solve for x, and finally interpret the result. He also reinforces basic algebraic steps: isolating x by moving constants across the equality sign and dividing by the appropriate coefficient when needed. He discusses how to explain steps verbally, such as why we subtract 28 from both sides to isolate x, and notes calculator tips for entering negative numbers (use the plus-minus key, if available, rather than a subtraction operation).

Worked Example 5: Mustard and Ketchup

A separate problem uses ketchup and mustard preferences among 100 people, with 12 people liking neither. Let x be the number who like both ketchup and mustard. The numbers who like only ketchup or only mustard are then (73 − x) and (54 − x) respectively, while 12 like neither. The total of these disjoint categories must equal 100, so
(73 - x) + (54 - x) + x + 12 = 100.
This simplifies to 139 - x = 100, hence x = 39.
Thus, 39 people like both condiments. A point of discussion in class is the placement of the x term (e.g., 73 − x versus x − 73). The correct phrasing is to place the subtractive term after the constant when describing the number of people in a single-category group, as long as the expression correctly represents the count of those people. The instructor also notes that a verification by summing all categories or by a complementary counting approach yields the same result.

Worked Example 6: Cell Phone and Tablet

In another example, students are asked to determine how many had both a cell phone and a tablet, given counts for each category and a total universal set. The instructor sets x as the number who have both. Then the number with cell phone only is 96 − x, and the number with tablet only is 74 − x. If the total number in the universal set is 120, then the equation becomes
120 = (96 - x) + x + (74 - x).
Solving yields x = 50, so 50 students have both a cell phone and a tablet. The class discusses the interpretation and consistency checks: the sum of the disjoint regions (cell only, tablet only, both) should equal the universal set when everyone has at least one of the two items.

Worked Example 7: J Homework and Set Builder

The instructor describes a homework problem (on page 20, items 1–5, and page 32, items 1–16) and explains a more advanced construction. In item six, a set J is defined by J = { j : j = 2x, x ∈ Z, −3 ≤ x ≤ 6 }. Here the variable x ranges over integers from −3 to 6, and each x yields a corresponding j value by j = 2x. Thus, A, the set of j’s, consists of the even integers generated by doubling each x in that interval:A = { -6, -4, -2, 0, 2, 4, 6, 8, 10, 12 }. The instructor notes that to determine the elements of A for any problem of this form, first enumerate the allowed x-values, apply the transformation j = 2x, and then list the resulting j-values.
The homework also includes a challenging j = 2x example to illustrate how the choice of x-domain affects the resulting set A. The teacher plans to discuss Question 6 in detail in class and encourages students to attempt the 21 questions for practice. He mentions that in some cases it is not necessary to draw the Venn diagram if the instruction does not require it. He also emphasizes that algebraic manipulation remains a core skill for these problems and that the objective is to learn to translate between verbal descriptions, set notation, and visual diagrams.

Recap: Equations, Notation, and Algebra Tips

The session reinforces several core ideas:

Universal set U frames the context for all operations; all sets A and B are subsets of U.
A ∪ B, A ∩ B, and A^c (and B^c) are the primary operations in Venn diagram problems.
The complement of A ∪ B is A^c ∩ B^c, and the complement of A ∩ B is A^c ∪ B^c (De Morgan’s laws).
When solving problems with overlaps, introducing a variable x for the intersection makes it easier to express the counts for A only, B only, and A ∩ B, and to set up a single equation using the total in U.
It is useful to write the equations clearly and to justify each algebraic step, such as why we subtract a constant from both sides to isolate x, and how to handle negative numbers in division or multiplication.
In practice, sometimes a quick verbal check or a small numerical verification (adding the components of the Venn diagram) helps confirm the result.

Homework Assignments and Study Guidance

The instructor assigns homework: pages 20 (problems 1–5) and page 32 (problems 1–16). For one of the problems (problem 6 on page 32), A is defined by A = { j : j = 2x, x ∈ Z, −3 ≤ x ≤ 6 }, which requires listing x-values first, computing j from j = 2x, and forming the set A. A student comment reminds that when solving these, it is acceptable to present the starting equation and the final answer, with the algebra steps shown as needed. The instructor notes that He will review problem 6 in class and that the 21 problems provide ample practice with translating between verbal statements and set notation, as well as with drawing and interpreting Venn diagrams.

Practical and Philosophical/Practical Implications

Throughout, the lecturer highlights practical implications of using Venn diagrams for set theory: they provide a visual means to organize data, reason about overlaps and exclusivity, and check consistency against the universal set. The discussion also touches on the compatibility of notation with technology (typing symbols like ∩, ∪) and the importance of being able to adapt representations (handwritten diagrams vs. digital format, multiple-choice or handwritten symbols). The exchange underscores the value of building intuition through concrete examples, then generalizing to abstract relationships such as subset relations, disjointness, and complements. Finally, the classroom dialogue models a collaborative learning process, where students ask questions, request clarifications, and work through problems step by step to reinforce understanding of Venn diagrams and set operations.