Set Note Form 4 2015

Set

1.A set is a collection of objects according to certain characteristics2.Theobjectsin a set are known aselements.3.Sets are usually denoted bycapital lettersand notation used for sets isbraces,{ }.Example:A= {1, 3, 5, 7, 9}

4.In set notation, the symbolmeansis an element oforbelongs toandmeansis not an element ofordoes not belong to.

Example 1:Given thatP= {factors of 15} andQ= {positive perfect squares less than 28}. By using the symbolor, complete each of the following:(a)5 ___ P (b)20 ___P (c)25 ___Q (d)8___Q

Solution:P= {1, 3, 5, 15},Q= {1, 4, 9, 16, 25}(a)5P5isanelementofsetP(b)20P20isnotanelementofsetP(c)25Q25isanelementofsetQ(d)8Q8isnotanelementofsetQ

Represent sets by using Venn diagram

5.A set can be represented by aVenn diagramusing closed geometry shapes such as circles, rectangles, triangles and etc.6.Adot to the leftof an object in a Venn diagram indicates that the object is anelementof the set.7.When a Venn diagram represents thenumberof elements in a set,no dotis placed to the left of the number.

Example 2:(a)Draw a Venn diagram to represent each of the following sets.(b)State the number of elements for each of the set.A= {2, 3, 5, 7}B= {k, m, r, t, y}

Solution:(a)

(b)n(A) = 4n(B) = 5

Determine whether a set is an empty set

8.A set withno elementsis called anempty setornull set. The symbolorempty braces, { }, denotes empty set.For example, if setAis an empty set, thenA= { } orA=andn(A) = 0.

9.IfB= {0} or {} does not denote thatBis an empty set.B= {0} means that there is an element 0 in setB.B= {} means that there is an element in setB.

Subsets1.If every element of a setAis also an element of a setB, then setAis calledsubsetof setB.

2.The symbolis used to denote is a subset of. Therefore, setAis a subset of setB. In set notation, it is written as AB.

Example:A= {11, 12, 13} andB= {10, 11, 12, 13, 14}Every element of setAis an element of setB. ThereforeAB.

3.ABcan be illustrated using Venn diagram as below:

4.The symbolis used to denote is not a subset of.5.Anempty setis asubsetof any set. For example,A6.A set is asubsetofitself. For example, BB

7.The number of subsets for a set withnelements is2n. For example, ifA= {3, 7} Son= 2, then number of subsets of setA= 22= 4 All the subsets of setAare { }, {3}, {7} and {3, 7}.

Universal Set1.Universal setis a set that contains all the elements under consideration.2.In set notation, the symboldenotes a universal set.

Example:Given that the universal set,= {whole numbers less than 9},A= {prime number} andB= {multiple of 4}.(a) List all the elements of setAand setB.(b) Illustrate the relationships between the following sets using Venn diagrams. (i) and A (ii) , A and B

Solution:(a)= {0, 1, 2, 3, 4, 5, 6, 7, 8} A= {2, 3, 5, 7} B= {4, 8}

(b)(i)

(b)(ii)

Complement of a Set

1. Thecomplementof setBis the set of all elements in the universal set,, which are not elements of setB, and isdenoted byB.

Example 1:If= {17, 18, 19, 20, 21, 22, 23} andB= {17, 20, 21} thenB = {18, 19, 22, 23}

2. The Venn diagram below shows therelationshipbetweenB,B and the universal set,.

The complement of setBis represented by the green colour shaded region inside the universal set,, but outside setB.

Intersection of Sets

1.Theintersectionof setPand setQ, denoted byPQis the set consisting of all elements common to setPand setQ.2.Theintersectionof setP, setQand setR, denoted byPQRis thesetconsisting of all elements common to setP, setQand setR.3.Represent the intersection of sets using Venn diagrams.

(a)PQ

(b)QP,thenPQ=Q

(c)PQ=,ThereisnointersectionbetweensetPandsetQ.

(d)PQR

Example 1:Given thatA= {3, 4, 5, 6, 7},B= {4, 5, 7, 8, 9, 12} andC= {3, 5, 7, 8, 9, 10}.(a)FindABC.(b)Draw a Venn diagram to representABC.

Solution:(a)ABC= {5, 7}(b)

4.Thecomplementof the intersection of two sets,PandQ, represented by(PQ), is a set that consists of all the elements of the universal set, , butnottheelements ofPQ.

5.The complement of set (PQ) is represented by the shaded region asshown in the Venn diagram.

Union of Sets

1.Theunionof setAand setB, denoted byABis the set consisting of all elements in setAor setBorboththe sets. The Venn diagram ofABis illustrated as below:

2.Theunionof setA, setBand setC, denoted byABCis the set consisting of all elements in setA, setBor setCorallthe three sets. The Venn diagram ofABCis illustrated as below:

Example 1:The Venn diagram below shows the number of elements in the universal set, , setP, setQandR.

Givenn(Q) =n(PR), findn().

Solution:n(Q) =n(PR)2x+ 6 + 1 + 5 = 2x+ 2x2x+ 12 = 4x2x= 12x= 6

n() = 2x+ 2x+x+ 7 + 6 + 1 + 5 = 5x+ 19 = 5(6) + 19 = 30 + 19 =49

Example 2:Diagram below is a Venn diagram showing the universal set, = {Form 3 students}, setA= {Students who play piano} and setB= {Students who play violin}.

Givenn() = 60,n(A) = 25,n(B) = 12 andn(AB) = 8, find the number of students who do not play the two instruments.

Solution:The students who do not play the two instruments are represented by the shaded region, (AB).

Number of students who do not play the two instruments=n(AB)= 60 17 8 4=31

Example 1:List all the subsets of setP= {r,s}.

Solution:There are 2 elements, so the number of subsets of setPis 2n= 22= 4.

SetP= {r,s}Thereforesubsets of setP= {r}, {s}, {r,s}, {}.

Example 2:

Diagram above shows a Venn diagram with the universal set, =QP.List all the subset of setP.

Solution:SetPhas 3 elements, so the number of subsets of setPis 2n= 23= 8.

SetP= {2, 3, 5}Thereforesubsets of setP= {}, {2}, {3}, {5}, {2, 3}, {2, 5}, {3, 5}, {2, 3, 5}.

Example 3:It is given that the universal set, = {x: 30 x< 42,xis an integer} andsetP= {x:xis a number such that the sum of it its two digits is an evennumber}.Find setP.

Solution: = {30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41}P= {31, 33, 35, 37, 39, 40}

ThereforeP= {30, 32, 34, 36, 38, 41}.

Example 4:Given that universal set = {x: 3