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Venn diagrams
SLIDE 1
Elements and sets
A set is a mathematical term for a collection of objects. To show that a collection forms a set, we use curly brackets.
The curly brackets indicate the set and the items inside the curly brackets are the elements of the set.
Examples
Sets may contain elements that are physical or concrete objects, but also elements that are non-physical or abstract objects. An arbitrary collection, or set, may be made from a mixture of concrete and abstract objects.
Elements are also called members of the set. We say “Africa is an element of the set of continents” or, equivalently, “Africa is a member of the set of continents”. Note. Another word used in mathematics for a set is a class.
SLIDE 2
? Write down the set of all planets.
? Write down the set of all even numbers less than 15.
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?
?
SLIDE 4
Finite and infinite sets
Sets can contain either finite or infinite members.
Example of a finite set
Example of an infinite set
SLIDE 5
Defining property
To form an infinite set we require a definition that specifies a defining property.
The defining property is that of being an even number
An arbitrary set may be made from any collection.
? Can a set be both infinite and arbitrary?
SLIDE 6
An arbitrary set cannot be infinite.
For a set to be an infinite collection (have infinite elements) we must specify through a definition giving a common property. So, an infinite set cannot be made from an arbitrary grouping of elements.
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A set may be defined explicitly or implicitly.
An explicit definition is one that states the common property of the set, or lists all the members of a finite set.
An implicit definition is a one that implies the common property of membership of a set by using dots. The reader is expected to continue the sequence in an “obvious” way.
Examples
The second example defines implicitly the set of all square numbers, because the reader is expected to continue the sequence that gives the elements in the obvious way.
Task
Use dots to define implicitly the set P above.
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?
SLIDE 9
The expression
defines a finite set A by listing its members a, b and c.
? Write down the finite set, B, that comprises as its members the elements a, b, d and e.
? Write down the finite set, C, that comprises as its members the elements that are in both A and B.
SLIDE 10
defines a finite set A by listing its members a, b and c.
? The finite set, B, that comprises as its members the elements a, b, d and e is
? The finite set, C, that comprises as its members the elements that are in both A and B is
SLIDE 11
Two sets are equal (or identical) if they have the same elements regardless of the order in which they are listed. Sets are collections of objects without regard to the order in which those objects occur.
.
These are the same set.
We use the symbol to indicate that two sets are not equal.
SLIDE 12
One of the following statements is true and one is false.
Which is true, and which is false?
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Set membership
We use the symbol ? to mean that an object is a member of a set.
This is read “Africa is a member of the set of continents”, or “Africa is an element of the set of continents”.
We use to mean that an object is not a member of a set.
This read, “3 is not an element of the set of all positive even numbers”.
? Express in words.
? Write “3 is not a square number” in symbols.
SLIDE 15
? ? “a is a member of the set comprising the elements a, b, c”.
? “3 is not a square number” ?
SLIDE 16
When we specify a set by means of a property, we use the notation
The colon inside the curly brackets is read “such that” and introduces the property that defines membership of the set. This is read, “The set planets is set of all object x such that x is a large body orbiting the Sun”.
Remarks
1. At this level the use of the colon to introduce the defining property of a set is rarely seen.
2. Sometimes, a vertical stroke is used instead of a colon.
This is an alternative notation and is also read, “such that”.
3. The expression “large body” used in the above definition of a planet is vague. (Astronomers in fact dispute whether Pluto should be regarded as a planet.)
SLIDE 17
? Translate “Saturn is a planet” using the symbol ?.
? Translate the following into words
SLIDE 18
? Translate “Saturn is a planet” using the symbol ?.
Solution
or
? Translate into words
Solution
5 is not an element of the set of all even numbers
or
5 is not an even number
SLIDE19
The null or empty set
We can also form a set with no objects.
This is called the null or empty set.
There is only one null set, because there is only one set with nothing in it! It is denoted by the symbol .
Note. An alternative notation for the null set is . Also a pair of curly brackets with nothing side can be used, .
State which of the following is the null set
? the set of all solutions to the equation , where
? the set of all integers such that
? the set of all integers such that
? the set of all fire-breathing dragons.
SLIDE 20
? the set of all solutions to the equation , where
? It is never permitted in mathematics to divide a non-zero number by 0, hence this set has no members and is null, .
? the set of all integers such that
? is not an integer, so this is the null set, .
? the set of all integers such that
? There is one number that satisfies this equation, and it is 1. So this is the set containing just the element, 1, and written .
? the set of all fire-breathing dragons
? Assuming that no fire-breathing dragons exist, then this is the null set, .
Remarks. It is easy to run into philosophical difficulties in the subject of set theory. Regarding ?, since dividing by 0 is never permitted, it could be said that is ill-defined, so does not define any set, not even the null one. Regarding ?, in mathematics we allow abstract objects to exist, so why should we not regard imaginary and metaphorical objects to exist? For example, the imaginary dragon Smaug would be a member of this set, and some people regard certain other people as “dragons”.
SLIDE 21
Domain of discourse
Every definition of a set implies a domain of discourse. This is the largest collection of objects to which the members of the set can belong. It will be clear from the context what the domain of discourse is. The domain of discourse may be represented by the symbol .
The domain of discourse is also called the universe (of discourse).
Advanced
If a domain of discourse is not defined, contradictions arise. The problem is analogous to the problem in algebra of dividing by 0. Dividing by 0 is not permitted in algebra, because you can derive a contradiction such as if you allow it. In set theory contradictions also arise when the largest set to which any object can belong is not given.
SLIDE 22
Sets of numbers
Very often the domain of discourse is a set of numbers. We have special symbols to denote these types of numbers.
Note. The set of natural numbers is also called the set of counting numbers. Natural number is the preferred more technical term for these numbers. They are whole positive numbers. It is a choice whether to include 0 in the set or not. Some do, so do not. Here we include 0.
SLIDE 23
Venn diagrams
A Venn diagram shows the relationship between sets. In a Venn diagram the domain of discourse is usually represented by a large rectangle, and the smaller sets contained within the domain of discourse by circles or other shapes.
Example
In this Venn diagram the domain of discourse is the set of all natural (or counting) numbers and the set A is a smaller set (called a subset) contained within this discourse. The use of rectangles and circles is arbitrary – any shape could be used. The position of the numbers within the circles is also arbitrary – they can be anywhere.
The set A can be written . This is read, “A is the set of all natural numbers n such that .”
SLIDE 24
Draw Venn diagrams to represent
? Domain of discourse is where
? Domain of discourse is .
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?
?
SLIDE 26
Order of a set
The number of elements in a set is called the order of the set and is denoted by .
Example. When , then .
Order of an infinite set
When a set is infinite its order is denoted by .
Example. When , then .
SLIDE 27
? The set P is defined by
.
Find .
? The set X is defined by .
Find .
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?
?
SLIDE 29
Biconditional
The symbol is called the biconditional. If two statements P and Q are such that if P is true, then Q is true, and vice-verse, if Q is true, then P is true, then we write .
This also means .
In words, the biconditional is expressed by the phrase, if, and only if. This is abbreviated to iff.
It is customary, but not essential, to write iff in bold.
Yet, another way of expressing the same idea is by the word “equivalent”. means P is equivalent to Q.
P if, and only if Q
P is equivalent to Q
have (in this context) the same meaning. The term “equivalent” is used in several ways in mathematics and logic, and this is one of its uses only.
SLIDE 30
Subsets
If B is a subset of the set A then all the elements of B are also elements of A. The symbol for subset is .
is read, “B is a subset of A”.
Venn diagrams make this relationship clear. In the above example, the domain of discourse is , and . B is a subset of A, and both are subsets of . All sets belonging to a domain of discourse are subsets of the domain of discourse.
Advanced
The definition of the subset property is uses the biconditional.
SLIDE 31
? List all the subsets of the set . In each case, state the order of the subset.
? How many subsets of A are there?
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?
?
SLIDE 33
Subsets of A
Observations
1. The set is a subset of itself.
Every set is a subset of itself. This follows from the definition of what a subset is for just says every element of A is an element of A.
2. The null set, , is a subset of A.
The null set is a subset of every set whatsoever. This follows from the definition of what a subset is. The null set contains no elements and so every element of the null set (that is no element) is also an element of the set A.
SLIDE 34
? is a subset of . Express this using the subset symbol .
? Let the domain of discourse, , be the first ten letters of the alphabet. Let and . Draw a Venn diagram to illustrate the relationships between , A and B.
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?
? , ,
SLIDE 36
The expression not an element of or not a member of is written .
Example
“15 is a member of A” (because 15 is divisible by 5)
“7 is not a member of A” (7 is not divisible by 5)
SLIDE 37
State whether the following are true or false
? ?
? ?
SLIDE 38
? ? true
? ? false
? ? true
? ? false
SLIDE 39
Complement of a set
The complement of a set A is the set of elements of (the domain of discourse, or domain) that are not elements of A.
The complement of A is denoted by the symbol .
The Venn diagram representation of a complement of a set A is
The shaded area represents the complement of A. Shading is sometimes added to a Venn diagram for the sake of clarity.
SLIDE 40
? Find the set
? Draw a Venn diagram showing the sets , A and .
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SLIDE 42
Union and intersection
The union of two sets, A and B is the set of elements that are elements of either A or B or both. Its symbol is .
The intersection of two sets, A and B is the set of elements that are elements of both A and B.
SLIDE 43
Find
? ?
? ?
In each case draw a Venn diagram to illustrate the solution.
SLIDE 44
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SLIDE 45
The figure shows a Venn diagram with the universal set and two subsets and . Write down the elements of
? ? A ? B
? ? ?
? ? ?
Draw the Venn diagram for each case.
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SLIDE 47
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Find the following sets.
? ?
? ?
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SLIDE 49
The numbers of elements in each region are shown in the following Venn diagram.
If , find
? ? ? ?
SLIDE 50
? ?
? ?
SLIDE 51
The entries of the Venn diagram show the number of elements in each region. If , find
? ? ?
? ? ?
? ?
SLIDE 52
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? ?
SLIDE 53 Typo in first line corrected – CHECK SLIDE
A class had 30 students everyone took at least one of the subjects History, Mathematics or Chemistry. 16 took History, 19 took Mathematics, and 18 took Chemistry. 10 students took both Mathematics and Chemistry. 11 students took Mathematics and History. 8 students took History and Chemistry.
? How many students took all three subjects?
? How many students took each subject?
SLIDE 54
Let x be the number of students taking all three subjects. Let s, r and t be the number of students taking only Chemistry, only Mathematics and only History respectively.
We are told that everyone took at least one subject, so
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