1. Foundations of Geometry

1.01 Introduction to the data cycle and Venn diagrams

1.02 Venn diagrams and sets

Lesson

Practice

1.03 Introduction to geometric notation

1.04 Line segments and constructions

1.05 Angles and constructions

Book a Demo

United States of America Virginia

Geometry

1.02 Venn diagrams and sets

Lesson

Practice

Ideas

Venn diagrams and sets

Venn diagrams and sets

A collection of items is called a set. There are many situations which we can describe and explore mathematically using sets, and to do so, we will need some new terminology.

Set

A collection of items, which are usually called elements. Sets are usually denoted using capital letters.

Example:

A=\left\{1,\,2,\,3,\,4,\,5, \ldots\right\}

Element

A single member of a set. Elements are usually denoted using lower case letters.

Subset

A set B is a subset of another set A, denoted B \subseteq A, if every element of B is also an element of A

Example:

If A=\left\{1,\,2,\,3,\,4,\,5\right\} and B=\left\{1,\,3,\,5\right\}, then B \subseteq A.

Proper subset

A subset B of A, where there is at least one element in A that is not in B.

Written as B \subset A.

Circle B inside circle A, then both are enclosed by a square. This is a representation of a subset.

A set can be described by listing its elements inside a pair of braces, and we call this set notation. For example, if the set A is "the set of positive integers smaller than nine", we can write this asA = \left\{1, 2, 3, 4, 5, 6, 7, 8\right\}

If there are too many elements to write out but there is a clear pattern to the elements, we can use three dots to indicate that a pattern continues. For example, if the set B is "the set of even whole numbers", we can write this as B = \left\{2, 4, 6, 8, 10, \ldots \right\}

A special set, called the empty set, is the set which contains no elements. It is usually represented by the symbol \emptyset, but can also be expressed in set notation as\emptyset = \left\{\right\}

Venn diagrams can be used to visually represent relationships between sets. Recall that a Venn diagram has four regions or subsets. These include:

a: The elements that belong only to the first set and not to the second set
b: The elements that belong to both sets
c: The elements that belong only to the second set and not to the first set
d: The elements that are included in the universal set but do not belong to either subset

Universal set

A set from which all elements in a problem can be found.

A Venn diagram showing two circles A and B. The entire rectangle is highlighted.

Negation

The negation of a set A, denoted A\rq or "not A" is the set of all elements of the universal set which are not elements of A.

This is sometimes called the complement.

A Venn diagram showing two circles A and B. Everything inside the rectangle except for the interior of circle A is highlighted.

Intersection

The intersection of two sets A and B, denoted {A \cap B}, is the set of all elements which belong to both A and B

This is sometimes called the conjunction.

A Venn diagram showing two circles A and B. The region that is inside both circles A and B is highlighted.

Union

The union of two sets A and B, denoted {A \cup B}, is the set of all elements which belong to either A or B

This is sometimes called the disconjunction.

A Venn diagram showing two circles A and B. The entirety of both circles A and B is highlighted.

A Venn diagram for Hockey and gymnastics. Hockey is shaded but not the intersection. Ask your teacher for more information.

We can describe each of the four regions using a combination of unions, intersection, and complements.

For example, the shaded region represents those who play Hockey \left(H\right), but don't do gymnastics \left(G\right). This means that in the set H\cap G\rq has 5 elements.

Exploration

The given Venn diagram shows the sets A, B, and C. Suppose that C is the set of all integers.

A venn diagram showing sets A,B and C. B is inside C. A overlaps both B and C.

With a partner, discuss these questions.

Describe a possible set B. Explain. Is there any other possible set?
Describe a possible set A. Explain.
Describe the elements in A\cap B\rq, the subset of elements in A, but not in B.

Examples

Example 1

P is the set of odd numbers between 2 and 16, and Q is the set given by Q = \left\{1, 2, 3, 5, 8, 13, 21\right\}.

Determine the set given by P \cap Q.

Worked Solution

Create a strategy

P \cap Q is the intersection of P and Q. That is, P \cap Q contains all of the elements that belong to both P and Q. In other words, we are looking for all the elements that are in Q and are also an odd number between 2 and 16.

Apply the idea

We can see that the odd elements of Q are 1, 3, 5, 13, and 21.

Of these, 3, 5, and 13 lie between 2 and 16 and are therefore also elements of P.

So, we have thatP \cap Q = \left\{3, 5, 13\right\}

Reflect and check

We can construct a Venn diagram to see the relationship between sets P and Q, particularly the elements they have in common.

Two overlapping circles within a rectangle, labeled P and Q, representing a Venn diagram. The left circle, P, contains the numbers 15, 11, 9, and 7 uniquely, while the right circle, Q, contains the numbers 1, 2, 8, and 21 uniquely. The overlapping area contains the numbers 3, 5, and 13, shaded in blue, suggesting shared elements between P and Q.

Example 2

In the universal setS = \left\{\text{square, triangle, rhombus, parallelogram, hexagon, circle, trapezoid, rectangle}\right\}the subset A is "quadrilaterals" and the subset B is "words beginning with the letter t".

Describe the set "not A" using words, then express it using set notation.

Worked Solution

Create a strategy

The keyword "not" indicates that we want to take the negation of A. In order to think about this, it might be useful to first list out the elements of A.

Apply the idea

The set A is the set of "quadrilaterals" from the universal set S. So we haveA = \left\{\text{square, rhombus, parallelogram, trapezoid, rectangle}\right\}

The negation of A is the set of all elements of S that are not in A. We can describe this as "the set of shapes which do not have exactly four sides". In set notation, this will beA\rq = \left\{\text{triangle, hexagon, circle}\right\}

Reflect and check

Two overlapping circles within a blue rectangle, labeled A and B, representing a Venn diagram. Circle A contains the names of shapes: Square, Rhombus, Parallelogram, and Rectangle. Circle B contains the names of shapes: Triangle and Hexagon. The overlapping area contains the name Trapezoid, indicating it as a shared element between A and B. The entire background within the rectangle is shaded blue.

This Venn diagram represents the universal set with A\rq highlighted.

Describe the set "B or not A" using words, then express it using set notation.

Worked Solution

Create a strategy

We determined the set "not A" in the first part of this question. The keyword "or" indicates that we want to take the union of this with set B. This means we need to include everything that is in set B as well as everything in "not A". This time it might be useful to list out the elements of B.

Apply the idea

The set B is the set of "words beginning with the letter t" from the universal set S. So we have B = \left\{\text{triangle, trapezoid}\right\}.

A\rq=\left\{\text{triangle, hexagon, circle}\right\} as we found in part (a).

The union of two sets will include any element that is in at least one of those sets. We can desribe the set "B or not A" as "the set of shapes that begin with the letter t or are not quadrilaterals." In set notation, this will beB \cup A\rq = \left\{\text{triangle, hexagon, circle, trapezoid}\right\}

Reflect and check

Taking the negation of A:

and combining it with set B:

Two overlapping circles within a blue rectangle, labeled A and B, represent a Venn diagram. Circle A contains the names of shapes: Square, Rhombus, Parallelogram, Rectangle, and Circle. Circle B contains the names of shapes: Triangle, Hexagon, and the overlapping section includes Trapezoid, shaded in blue, indicating it as a shared element between A and B. The rest of circle B is also shaded in blue.

We get the Venn diagram:

Two overlapping circles within a blue rectangle, labeled A and B, form a Venn diagram. Circle A contains the names of shapes: Square, Rhombus, Parallelogram, Rectangle, and Circle. Circle B contains the names of shapes: Triangle and Hexagon. The overlapping area contains the name Trapezoid, indicating it as a shared element between A and B. The entire background within the rectangle is shaded blue.

Notice that the element "triangle" appears in both B and A\rq. This element is still included in the union, since appearing in both sets means that it appears in at least one of the two sets.

We only need to list it once; we do not need to write the element twice in the union even though it appears in both sets.

Example 3

Consider the sets shown in the Venn diagram:

A Venn diagram showing circles A and B and a number of elements. See your teacher for more information.

Identify the elements in the universal set.

Worked Solution

Create a strategy

The universal set contains all the possible elements that can appear in any set in the context. For a Venn diagram, this includes any element within the outer rectangle (whether or not it is inside any circles).

Apply the idea

In this case, every number from 1 to 20 appears exactly once, so the sample space isS=\left\{1, 2, 3, \ldots, 19, 20\right\}

Reflect and check

In any universal set, the sets A and B are subsets of the universal set. Using geometric notation, we can write A\subseteq S (A is a subset of S) and B\subseteq S (B is a subset of S).

There are many other subsets in a universal set such as A\rq and A\cup A\rq and A\cap B, to name a few.

Indicate the region which represents A \cap B\rq on the diagram, and determine the elements of this set.

Worked Solution

Create a strategy

B\rq is the negation of B which is the region outside of circle B.

Two overlapping circles within a blue square, labeled A and B respectively. The circles are drawn in a dark outline with the overlapping section clearly visible, indicating the common area between sets A and B. The entire background of the square is shaded blue.

We then want to take the intersection of that with set A.

A Venn diagram showing two circles A and B. Circle A is highlighted.

The intersection will be the region where the shading overlaps.

Apply the idea

We want to include everything that is inside circle A but not inside circle B:

A Venn diagram showing circles A and B and a number of elements, with one region highlighted. See your teacher for more information.

In set notation, this isA \cap B\rq = \left\{1, 4, 16\right\}

Reflect and check

In part (a), we found that the universal set for this Venn diagram is "all integers from 1 to 20 inclusive".

Within this universal set, we can describe set A as "the set of square numbers", and we can describe set B as "integers between 6 and 12 inclusive".

Within the universal set, we could also describe the set A \cap B\rq as "square numbers that are not between 6 and 12" The set A \cap B\rq is a subset of the universal set.

Write the set notation that represents the set \{6, \,7, \,8, \,10 , \,11, \,12\}.

Worked Solution

Create a strategy

We can shade the area of the Venn diagram where these numberse are found, describe the set in words, and then use set notation to describe the set.

Apply the idea

A Venn diagram showing circles A and B and a number of elements, with region of B highlighted. See your teacher for more information.

These numbers are found in this shaded area of the Venn diagram.

In words, we can describe this as in set B, but not in set A.

In set notation we can say B \cap A\rq.

Reflect and check

We could also say A\rq \cap B.

Example 4

A group of students were asked why they skipped breakfast. The two reasons given were that they were "not hungry" and they were "too busy".

A Venn diagram with 2 sets: not hungry and too busy that overlap. Ask your teacher for more information.

How many of the students skipped breakfast because of exactly one reason?

Worked Solution

Create a strategy

Add the number of students that are in the "Not hungry" circle or in the "Too busy" circle but not in both.

A Venn diagram with 2 sets that overlap. The circles are shaded but not the intersection.

Apply the idea

\displaystyle \text{Students with one reason}	\displaystyle =	\displaystyle 8 + 13	Add the numbers
	\displaystyle =	\displaystyle 21	Evaluate

Let N represent the set of students who skipped breakfast because they were not hungry. Let B represent the set of students who were too busy.

Use set notation to describe the region which contains all students who skipped breakfast.

Worked Solution

Create a strategy

Set notation means to use symbols like \cap, \cup, and \rq.

Apply the idea

The students that skipped breakfast are contained within the two circles.

This is the union or N \cup B.

Reflect and check

Since there isn't a number outside of the two circles, we can assume that there aren't any students who skipped breakfast for another reason.

Use set notation to describe the region which contains the students who only skipped breakfast because they were too busy.

Worked Solution

Create a strategy

The word "only" means excluding other options. We can rewrite this in words and then write using set notation.

Apply the idea

Since we want the students who only said "too busy," we should not include the intersection.

We can think of this as those who are in B, but not in N.

This is the set B \cap N\rq.

Reflect and check

This subset would have 13 students in it. While the subset B of "all students who gave too busy as a reason" would have 24 students in it.

Example 5

One hundred students in a school are asked about the subjects that they study. 58 of them are studying both math and science, 70 are studying math, and 23 are not studying math nor science.

Construct a Venn diagram that represents the given information.

Worked Solution

Create a strategy

There are two sets: students who study math and students who study science. We are also considering their negations: students who do not study math and students who do not study science.

Apply the idea

The two sets will represent the two circles in the Venn diagram.

Two overlapping circles within a rectangle, labeled Math for the left circle and Science for the right circle. The circles are drawn with a dark outline to denote the distinct and shared areas of these two academic subjects. The background is white, emphasizing the clear, distinct lines of the Venn diagram.

We know that 58 students study both math and science. This is the intersection of the sets, so we can put this number in the center section where the two circles overlap:

Two overlapping circles within a rectangle, labeled Math for the left circle and Science for the right circle. The number 58 is placed in the overlapping section, indicating a shared attribute or quantity related to both subjects. The diagram is outlined in dark lines against a white background.

We also know that 70 students are studying math, but this includes the 58 that are studying science. So there are 70-58=12 that are studying math, but not science. These students belong in the left circle but not the right circle:

Two overlapping circles within a rectangle, labeled Math for the left circle and Science for the right circle. The left circle contains the number 12, exclusive to Math, while the number 58 is placed in the overlapping section, indicating a shared attribute or quantity related to both Math and Science. The diagram is outlined in dark lines against a white background.

Next, we have also been told that 23 students are not studying either of these subjects. This number should be in the rectangle that represents the whole set, but not in either of the circles:

To complete the Venn diagram, we need to determine how many students are studying only science. We can use the fact that there are 100 students in total to find how many study only science. Since all four regions sum to 100, we can subtract the three that are filled in:100-23 - 12 - 58 = 7

This number goes in the right side of the circle for science only and completes the Venn diagram.

Venn diagram showing Math (12), Science (7), intersection (58) and universal set (23).

One student is chosen at random. Determine the probability of randomly selecting a student that is studying science.

Worked Solution

Create a strategy

To calculate probability, we can use that: \text{Probability of an event}=\dfrac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}

There are 100 students in total, so when calculating probability this will the the number of possible outcomes. We need to find the number of favorable outcomes.

Apply the idea

Two overlapping circles within a rectangle, labeled Math for the left circle and Science for the right circle, which is shaded in blue. The left circle contains the number 23 and 12, with 23 outside the overlapping area and 12 within it, exclusive to Math. The right circle contains the number 7, exclusive to Science, and shares the number 58 with Math in the overlapping section. The diagram is outlined in dark lines against a white background.

This shaded region shows all students who are studying science, the favorable outcome. The number of students in this circle is 58+7=65.

\displaystyle \text{Probability of an event}	\displaystyle =	\displaystyle \dfrac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}
	\displaystyle =	\displaystyle \dfrac{65}{100}
	\displaystyle =	\displaystyle 65\%

The probability of a randomly selecting a student studying science is 65\%.

Let M represent students who study Math and let S represent students who study Science.

One student is chosen at random. Determine the probability of randomly selecting a student in the set \left(M\cup S\right)\rq.

Worked Solution

Create a strategy

We can translate this set notation into words and then use the Venn diagram to find the number of favorable outcomes.

Apply the idea

\left(M\cup S\right)\rq is the complement or negation of M\cup S, which is the union of the two sets.

A venn diagram showing Math(12), Science(7), intersection (58) and universal set (23). All regions of Math and Science, and also their intersection are shaded.

M\cup S can be shown by shading the union and represents all students who study math or science.

A venn diagram showing Math(12), Science(7), intersection (58) and universal set (23). The universal set is shaded.

\left(M\cup S\right)\rq can be shown by shading everything outside of the union and represents all students who study neither math nor science.

There are 23 students in the set \left(M\cup S\right)\rq.

\displaystyle \text{Probability of an event}	\displaystyle =	\displaystyle \dfrac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}
	\displaystyle =	\displaystyle \dfrac{23}{100}
	\displaystyle =	\displaystyle 23\%

The probability of a randomly selecting a student in \left(M\cup S\right)\rq who is not studying math nor science is 23\%.

Reflect and check

Notice that:

\displaystyle P\left(\left(M\cup S\right)\rq\right)+P\left(M\cup S\right)	\displaystyle =	\displaystyle \dfrac{23}{100}+\dfrac{12+58+7}{100}
	\displaystyle =	\displaystyle \dfrac{23}{100}+\dfrac{77}{100}
	\displaystyle =	\displaystyle \dfrac{100}{100}
	\displaystyle =	\displaystyle 100\%

This means that the probability of a set and its complement is 1 or 100\%. These two sets make up the universal set, so the favorable outcomes are the same as the possible outcomes.

Explain why P\left(M\right)+P\left(M\rq\right)=1.

Worked Solution

Create a strategy

Remember that M is the set of students who are studying math and M\rq is the set of students who are not studying math.

Apply the idea

A student in the universal set must be in M, studying math, or in M\rq, not studying math. There are no other possibilities. For this particular case:

\displaystyle P\left(M\right)+P\left(M\rq\right)	\displaystyle =	\displaystyle \dfrac{12+58}{100}+\dfrac{23+7}{100}
	\displaystyle =	\displaystyle \dfrac{70}{100}+\dfrac{30}{100}
	\displaystyle =	\displaystyle \dfrac{100}{100}
	\displaystyle =	\displaystyle 1

Reflect and check

We can make a connection to logic notation and the idea that p \lor \sim p is always true. This is because if p is true then one of the whole statement is true and if p is false, then \sim p is true, so the whole statement is true.

Idea summary

When working with sets, we often want to find new subsets of a universal set.

Where the two subsets overlap is called the intersection, is written as A \cap B and contains the elements in both A and B.

Everything in the two subsets is called the union, is written as A \cup B, and contains the elements in either A or B.

Everything outside one of subsets is called the negation or complement, is written as A \rq, and contains the elements not in A.

A venn diagram showing Circle A and circle B, overlapped, then enclosed within a rectangle.

Outcomes

G.RLT.1

The student will translate logic statements, identify conditional statements, and use and interpret Venn diagrams.

G.RLT.1c

Use Venn diagrams to represent set relationships, including union, intersection, subset, and negation.

G.RLT.1d

Interpret Venn diagrams, including those representing contextual situations.

1.02 Venn diagrams and sets

Ideas

Venn diagrams and sets

Exploration

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Outcomes

G.RLT.1

G.RLT.1c

G.RLT.1d

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