Set Theory Topic 2: Sets and Set Notation

I can provide examples of the empty set, disjoint sets, subsets, and universal sets in context, and explain the reasoning. I can organize information such as collected data and number properties, using graphic organizers, and explain the reasoning. I can determine the elements in the complement of two sets.

Explore… Consider the following diagram that looks at one possible way of grouping the letters of the alphabet.

Explore… 1. How are the letters grouped? 2. Why is the letter Y in an overlapping section of the diagram? 3. How many letters are in the first circle? How many letters are in the second circle? When you add these two numbers together, is the answer the same as the number of letters in the alphabet? Why or why not? 4. Are there any other ways that you could group these letters? Try this explore activity in your workbook before looking at the answers on the next slide.

You should notice… 1. How are the letters grouped? They are grouped according to their classification of vowel or consonant. 2. Why is the letter Y in an overlapping section of the diagram? Y is considered to be both a consonant and a vowel. 3. How many letters are in the first circle? How many letters are in the second circle? When you add these two numbers together, is the answer the same as the number of letters in the alphabet? Why or why not? There are 6 items in circle A and 21 in circle B. This adds to 27, since Y is counted twice. 4. Are there any other ways that you could group these letters? Yes. For example, according to whether or not the letter contains a straight line.

Information A set is a collection of distinguishable objects. The objects in a set are called elements. A universal set contains all the elements in a particular context. It is also called the sample space. For example, the universal set of digits is D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. A subset is a smaller set whose elements all belong to a bigger set. The complement is the leftover elements of a universal set that do not belong to any subset.

Information Disjoint sets are two or more sets that have no elements in common. An empty set is a set that has no elements in it. A Venn diagram is a graphical way to show all the possible relations between a number of sets. Numbers in a Venn diagram can represent the number of elements in a particular set, or be the numbers themselves.

Example 1 Canada’s provinces and territories can be categorized using sets. Using sets to categorize items

Example 1 a) List the elements of the universal set, C, of Canadian provinces and territories. Sets are defined using brackets. For example: U = {1, 2, 3} C = {BC, AB, SK, MT, ON, QC, NL, NS, NB, PEI, YU, NWT, NVT}

Example 1 b) List the set of 4 western provinces and 3 western territories, W, in set notation. How is W related to C? W = {BC, AB, SK, MT, YU, NWT, NVT} To show that A is a subset of U we write: (W is a subset of C).

Example 1 c) List T, the set of territories. Is T a subset of another set? T = {YU, NVT, NWT} T, the Canadian territories is a subset of W and also a subset of C. This is because all of the territories in Canada are considered to be in the subset W.

Example 1 d) The complement W is W’. i. Describe what W’ contains ii. Write W’ in set notation. iii. Explain what represents in the Venn diagram from part c). i. W’ contains all Canadian provinces and territories that are not considered to be western. ii. W’ = {ON, QC, NL, PEI, NS, NB} iii. W’ is all the elements in C that are not in W.

Example 1 e) The set of eastern provinces is E = {NL, PEI, NS, NB, QC, ON}. Is E equal to W’? Explain. Yes. E is equal to W’, since they contain all of the same elements.

Example 1 f) Draw a Venn diagram representing C, W, T, and E. List all elements in the appropriate circle(s). C W BC SK NVT AB YU MT NWT T E NL PEI NS NB QC ON

Example 1 g) Explain why you can represent the set of Canadian provinces south of Mexico by the empty set. There are no Canadian provinces South of Mexico. The empty set is written as { } or Ø. h) Consider the sets C, W, E, W’, and T. List all the pairs of disjoint sets. Disjoint sets include W and E, W and W’, T and E, T and W’. Canada Mexico

Example 2 A triangular number, such as 1, 3, 6, or 10, can be represented as a triangular array. a) Determine a pattern you can use to find any triangular number. Determining the number of elements in sets For each new triangle, add a number 1 more than the previous added number.

Example 2 b) Determine the following sets: i. U = {natural numbers from 1 to 21 inclusive} ii. T = {triangular numbers from 1 to 21 inclusive} iii. E = {even triangular numbers from 1 to 21 inclusive} iv. O = {odd triangular numbers from 1 to 21 inclusive} v. T’ = {non-triangular numbers from 1 to 21 inclusive} Before you can do all of this, continue the pattern to determine all subsequent triangular numbers: 1, 3, 6, 10, 15, 21 Try this example in your workbook before looking at the answers on the next slide.

Example 2 b) i. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} ii. T = {1, 3, 6, 10, 15, 21} iii. E = {6, 10} iv. O = {1, 3, 15, 21} v. T’ = {2, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20} 1, 3, 6, 10, 15, 21

Example 2 c) Using set notation, how many elements are in sets U, T, and E? d) Using set notation and your answers from part c), calculate the number of elements in sets O and T’. In set notation, the number of elements of set X is written as n(x). n(U) = 21n(T) = 6n(E) = 2 n(O) = 6 – 2 = 4 n(T’) = 21 – 6 = 15

Example 2 e) Draw a Venn diagram to represent the sets in (b). O T U E T’

Example 2 f) Using set notation, list the sets that are subsets of other sets. O T U E T’

Example 2 g) Why is the set of odd triangular numbers from 1 to 21 defined as set O instead of being defined as E’? h) Consider the set of all natural numbers. How many numbers are triangular numbers: Is this set finite or infinite? An infinite set, such as the set of natural numbers, is written as N = {1, 2, 3…}. E’ would include all natural numbers between 1 and 21 that are NOT even and triangular. O includes only the number from 1 to 21 that are odd triangular numbers. The triangular numbers continue on forever, so it would be considered an infinite set.

Example 3 Jarrod and Luke rescue homeless animals. They currently have cats, dogs, rabbits, hamsters, parrots, lovebirds, iguanas, and snakes. The boys need to design a web page to help them advertise to find homes for the animals. They first must organize the animals, so they grouped them into the following sets: A = {all the animals that are available} W = {warm-blooded animals} C = {cold-blooded animals} Describing the relationships between sets

Example 3 a) Using set notation, list the elements of W and C. b) Two possible subsets of W are M, the set of mammals, and B, the set of birds. Using set notation, list the elements of these subsets. c) Use a Venn diagram to represent A, W, C, and the two subsets of W. d) Name any disjoint sets. e) Use set notation to show which sets are subsets of one another. f) Jarrod said that the set of fur-bearing animals could form one subset. Name another set of animals that is equal to this subset. g) How else might you categorize the animals into sets and subsets? Try this example in your workbook before looking at the answers on the next slide.

Example 3: Solution a) Using set notation, list the elements of W and C. b) Two possible subsets of W are M, the set of mammals, and B, the set of birds. Using set notation, list the elements of these subsets. W = {cats, dogs, rabbits, hamsters, parrots, lovebirds} C ={iguanas, snakes} M = {cats, dogs, rabbits, hamsters} B ={parrots, lovebirds}

Example 3: Solution c) Use a Venn diagram to represent A, W, C, and the two subsets of W. B W A M C

Example 3: Solution d) Name any disjoint sets. e) Use set notation to show which sets are subsets of one another. f) Jarrod said that the set of fur-bearing animals could form one subset. Name another set of animals that is equal to this subset. g) How else might you categorize the animals into sets and subsets? M and B, M and C, B and C, W and C mammals Reptiles, birds and mammals, nocturnal and not nocturnal.

Need to Know A set is a collection of distinguishable objects called elements. A set can be represented by: listing the elements in set notation; for example, A = {1, 2, 3, 4, 5} using words, for example, A = {all integers greater than 0 and less than 6} In set notation, the number of elements of set X is written as: n(X).

Need to Know A universal set, also known as the sample space, contains all the elements in a particular context. A universal set can be split into subsets, often in more than one way. For example, if set A is a subset of set U, the notation is used. Venn diagrams can be used to show how sets and their subsets are related. An empty set is a set that has no elements in it and is denoted as { }.

Need to Know Disjoint sets are two or more sets that have no elements in common. The complement of a set contains all elements that do not belong to the set. The sum of the number of elements in a set and its complement is equal to the number of elements in the universal set n(A)+n(A)’ = n(U). You’re ready! Try the homework from this section.