9. P is the set of all perfect squares bigger than 1 and
less than 50.
(i) Is 10 EP?
(ii) List all the elements of P in set notation.
Answers
Answer:
SETS AND VENN DIAGRAMS
Number and Algebra : Module 1Years : 7-8
June 2011
PDF Version of module
Assumed knowledge
Motivation
Content
Describing and naming sets
Subsets and Venn diagrams
Complements, intersections and unions
Solving problems using a Venn diagram
Links Forward
Sets and probability
History and applications
Answers to Exercises
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ASSUMED KNOWLEDGE
Addition and subtraction of whole numbers.
Familiarity with the English words
‘and’, ‘or’, ‘not’, ‘all’, ‘if…then’.
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MOTIVATION
In all sorts of situations we classify objects into sets of similar objects and count them. This procedure is the most basic motivation for learning the whole numbers and learning how to add and subtract them.
Such counting quickly throws up situations that may at first seem contradictory.
‘Last June, there were 15 windy days and 20 rainy days, yet 5 days were neither windy nor rainy.’
How can this be, when June only has 30 days? A Venn diagram, and the language of sets, easily sorts this out.
Let W be the set of windy days,
and R be the set of rainy days.
Let E be the set of days in June.
Then W and R; together have size 25, so
the overlap between W and R is 10.; The Venn diagram opposite displays; the whole situation.
The purpose of this module is to introduce language for talking about sets, and some notation for setting out calculations, so that counting problems such as this can be sorted out. The Venn diagram makes the situation easy to visualise.
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CONTENT
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DESCRIBING AND NAMING SETS
A set is just a collection of objects, but we need some new words and symbols and diagrams to be able to talk sensibly about sets.
In our ordinary language, we try to make sense of the world we live in by classifying collections of things. English has many words for such collections. For example, we speak of ‘a flock of birds’, ‘a herd of cattle’, ‘a swarm of bees’ and ‘a colony of ants’.
We do a similar thing in mathematics, and classify numbers, geometrical figures and other things into collections that we call sets. The objects in these sets are called the elements of the set.
Describing a set
A set can be described by listing all of its elements. For example,
S = { 1, 3, 5, 7, 9 },
which we read as ‘S is the set whose elements are 1, 3, 5, 7 and 9’. The five elements of the set are separated by commas, and the list is enclosed between curly brackets.
A set can also be described by writing a description of its elements between curly brackets. Thus the set S above can also be written as
S = { odd whole numbers less than 10 },
which we read as ‘S is the set of odd whole numbers less than 10’.
A set must be well defined. This means that our description of the elements of a set is clear and unambiguous. For example, { tall people } is not a set, because people tend to disagree about what ‘tall’ means. An example of a well-defined set is
T = { letters in the English alphabet }.
Equal sets
Two sets are called equal if they have exactly the same elements. Thus following the usual convention that ‘y’ is not a vowel,
{ vowels in the English alphabet } = { a, e, i, o, u }
On the other hand, the sets { 1, 3, 5 } and { 1, 2, 3 } are not equal, because they have different elements. This is written as
{ 1, 3, 5 } ≠ { 1, 2, 3 }.
The order in which the elements are written between the curly brackets does not matter at all. For example,
{ 1, 3, 5, 7, 9 } = { 3, 9, 7, 5, 1 } = { 5, 9, 1, 3, 7 }.
If an element is listed more than once, it is only counted once. For example,
{ a, a, b } = { a, b }.
The set { a, a, b } has only the two elements a and b. The second mention of a is an unnecessary repetition and can be ignored. It is normally considered poor notation to list an element more than once.
The symbols ∈ and ∉
The phrases ‘is an element of’ and ‘is not an element of’ occur so often in discussing sets that the special symbols ∈ and ∉ are used for them. For example, if A = { 3, 4, 5, 6 }, then
3 ∈ A (Read this as ‘3 is an element of the set A’.)
8 ∉ A (Read this as ‘8 is not an element of the set A’.)
Describing and naming sets
A set is a collection of objects, called the elements of the set.
A set must be well defined, meaning that its elements can be described and
listed without ambiguity. For example:
{ 1, 3, 5 } and { letters of the English alphabet }.
Two sets are called equal if they have exactly the same elements.
The order is irrelevant.
Any repetition of an element is ignored.
If a is an element of a set S, we write a ∈ S.
If b is not an element of a set S, we write b ∉ S.
Step-by-step explanation:
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Answer:
no 10 is not perfect square
p= 4,9,16,25,36,49