Home › SPM Maths › Archive for Sets

Sets

3.4 SPM Practice (Long Questions)

Posted on by

Question 9 (3 marks):
The Venn diagram in the answer space shows set P, set Q and set R such that the universal set ξ = P U Q U R.
On the diagram in the answer space, shade the set
(a) P’,
(b) (P ∩ Q) U R

Answer:



Solution:
(a)



(b)





Question 10 (3 marks):
(a) It is given that set E = {perfect square numbers} and set F = {9, 16, 25}.
Complete the Venn diagram in the answer space to show the relationship between set E and set F.

Answer:


(b) The Venn diagram in Diagram 1 shows the sets X, set Y and set Z.
The universal set, ξ = X U Y U Z.

Diagram 1

State the relationship represented by the shaded region between sets X, set Y and set Z.



Solution:
(a)


(b)


Hence, relationship represented by the shaded region between sets X, set Y and set Z are (X ∩ Y) ∪ Z.


3.4 SPM Practice (Long Questions)

Posted on by

Question 7:
The Venn diagrams in the answer space shows sets P, Q and R such that the universal set, ξ=P∪Q∪R .
On the diagrams in the answer space, shade set
(a) P ∩ Q,
(b) P ' ∩Q∪R


Solution:
(a) 
P ∩ Q


(b) 
P ' ∩Q∪R



Question 8:
The Venn diagrams in the answer space shows sets P, Q and R such that the universal set, ξ=P∪Q∪R .
On the diagrams in the answer space, shade set
(a) Q ' ∩R
(b) ( Q∪R )∩ P '



Solution:
(a) 
Q ' ∩R


(b) 
( Q∪R )∩ P '



3.4 SPM Practice (Long Questions)

Posted on by

Question 5:
The Venn diagrams in the answer space shows sets P, Q and R such that the universal set, ξ=P∪Q∪R
On the diagrams in the answer space, shade
(a) P ∩ Q,
(b) Q∩( P ' ∪R).

Solution:
(a) 
P ∩ Q

(b)
Q∩( P ' ∪R).




Question 6:
The Venn diagrams in the answer space shows sets P, Q and R such that the universal set, ξ=P∪Q∪R
On the diagrams in the answer space, shade
(a) P ∩ Q,
(b) P∩ (Q∪R) ' .

Solution:
(a)
P ∩ Q


(b)
P∩ (Q∪R) ' .



3.4 SPM Practice (Long Questions)

Posted on by

Question 1:
The Venn diagrams in the answer space shows sets X, Y and Z such that the universal set, ξ = X ∪ Y ∪ Z
On the diagrams in the answer space, shade
( a ) X' ∩ Y , ( b ) ( X ∪ Y' ) ∩ Z


Solution:
(a) 
• X’ ∩ Y means the intersection of the region outside X with the region Y.

 
 
(b)
• Find the region of (X υ Y’) first.
• (X υ Y’) means the union of the region X and the region outside Y.
• The region then intersects with region Z to give the result of (X υ Y’) ∩ Z.





Question 2:
The Venn diagrams in the answer space shows sets P, Q and R such that the universal set, ξ = P υ Q υ R
On the diagrams in the answer space, shade
(a) Q ∩ R,
(b) (P’ ∩ R) υ Q.

Solution:
(a) 
• Q ∩ R means the intersection of the region Q and the region R.



(b)
• Find the region of (P’ ∩ R) first.
• (P’ ∩ R) means the region that is outside P and is inside R.
• The union of this region with region Q give the result of (P’ ∩ R) υ Q.



3.4.2 Sets, SPM Paper 1 (Short Questions 5 – 7)

Posted on by

3.4.2 Sets, SPM Practice Paper 1 (Short Questions)

Question 5:
Diagram below shows a Venn diagram with the number of elements of set P, set Q and set R.

It is given that the universal set, ξ=P∪Q∪R and n( Q' )=n( Q∩R ).
Find the value of x.

Solution:
n(Q)’ = n(Q ∩ R)
3 + 8 + 5 = x– 3 + 9
16 = x + 6
x = 10


Question 6:
Diagram below is a Venn diagram showing the number of quiz participants in set P, set Q and set R.
It is given that the universal set, ξ = P ∪ Q ∪ R , set P = {Science quiz participants}, set Q = {Mathematics quiz participants} and set R = {History quiz participants}.

If the number of participants who participate in only one quiz is 76, find the total number of the participants.

Solution:
Number of participants who participate in only one quiz = 76
(5x – 2) + (x + 6) + (2x + 8) = 76
8x + 12 = 76
8x = 64
x = 8
Total number of the participants
= 76 + 7 + 4 + 5 + 3(8)
= 116


Question 7:
Diagram below is a Venn diagram showing the number of students in set K, set L and set M.
It is given that the universal set, ξ = K ∪ L ∪ M , set K = {Karate Club}, set L = {Life Guards Club} and set M = {Martial Arts Club}.

If the number of students who join both the Life Guards Club and the Martial Arts Club is 8, find the number of students who join only two clubs.

Solution:
Number of students who join both the Life Guards Club and the Martial Arts Club = n(L ∩ M) = 2 + 2x
2 + 2x = 8
2x = 6
x = 3
Number of students who join only two clubs
= x + 4 + 2x
= 3 + 4 + 2(3)
= 13

3.4.1 Sets, SPM Paper 1 (Short Questions 1 – 4)

Posted on by

Question 1:
List all the subsets of set P = {r, s}.
 
Solution:
There are 2 elements, so the number of subsets of set P is 2n = 22= 4.
Set P = {r, s}
Therefore subsets of set P = {r}, {s}, {r, s}, { }.


Question 2:
 
Diagram above shows a Venn diagram with the universal set, ξ = Q υ P.
List all the subset of set P.
 
Solution:
Set P has 3 elements, so the number of subsets of set P is 2n = 23 = 8.
Set P = {2, 3, 5}
Therefore subsets of set P = { }, {2}, {3}, {5}, {2, 3}, {2, 5}, {3, 5}, {2, 3, 5}.


Question 3:
It is given that the universal set, ξ = {x: 30 ≤ x < 42, x is an integer} and set P = {x: x is a number such that the sum of it its two digits is an even number}.
Find set P’.
 
Solution:
ξ = {30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41}
P = {31, 33, 35, 37, 39, 40}
Therefore P’ = {30, 32, 34, 36, 38, 41}.


Question 4:
Given that universal set ξ = {x : 3 < x ≤ 16, x  is an integer},
Set A = {4, 11, 13, 16},
Set B = {x : x is an odd number} and
Set C = {x : x is a multiple of 3}.

The elements of the set (A υ C)’ ∩ B are

Solution:
ξ = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}
A = {4, 11, 13, 16}
B = {5, 7, 9, 11, 13, 15}
C = {6, 9, 12, 15}
(A υ C)’ = {5, 7, 8, 10, 14}
Therefore (A υ C)’ ∩ B = {5, 7}.

3.3 Operations on Sets

Posted on by
3.3b Union of Sets (Part 2)
Example 1:
The Venn diagram below shows the number of elements in the universal set, ξ, set P, set Q and R.


Given n(Q) = n(P υ R)’, find n(ξ).
 
Solution:
n(Q) = n (P υ R)’
2x + 6 + 1 + 5 = 2x + 2x
2x + 12 = 4x
2x = 12
x = 6
 
n(ξ) = 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, ξ = {Form 3 students}, set A = {Students who play piano} and set B = {Students who play violin}.
 

 
Given n(ξ) = 60, n(A) = 25, n(B) = 12 and n(A ∩ B) = 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, (A υ B)’.

 
Number of students who do not play the two instruments
= n (A υ B)’
= 60 – 17 – 8 – 4
= 31
 

3.3a Intersection of Sets (Part 2)

Posted on by
3.3a Intersection of Sets
Example 1:
Given that A= {3, 4, 5, 6, 7}, B = {4, 5, 7, 8, 9, 12} and C = {3, 5, 7, 8, 9, 10}.
(a) Find A∩B∩C.
(b) Draw a Venn diagram to represent A∩B∩C.
 
Solution:
(a)  A∩B∩C= {5, 7}

(b)


 

4. The complement of the intersection of two sets, P and Q, represented by (P ∩ Q)’, is a set that consists of all the elements of the universal set, ξ, but not the elements of P ∩ Q.

5. The complement of set (P ∩ Q)’ is represented by the shaded region as shown in the Venn diagram.


 
 
 
 

3.4 SPM Practice (Long Questions)

Posted on by

Question 3:
(a) The Venn diagrams in the answer space shows sets P and Q such that the universal set, ξ = P υ Q.
  Shade the set P ∩ Q.
(b) The Venn diagrams in the answer space shows sets X and Y and Z, such that the universal set, ξ = X υ Y υ Z.
  Shade the set (X υ Z) ∩ Y.

Solution:
(a)

• P ∩ Q means the intersection of the region P and the region Q.

(b)


• (X υ Z) means the union of the region X and the region Z.
• The region then intersects with region Y to give the result (X υ Z) ∩ Y.





Question 4:
The Venn diagrams in the answer space shows sets P, Q and R such that the universal set, ξ = P υ Q υ R
On the diagrams in the answer space, shade
(a) P ∩ R’,
(b) P’υ (Q ∩ R).

Solution:
(a)
P ∩ R’
(b)
P’υ (Q ∩ R)


3.1 Set

Posted on by
3.1 Set
1. A set is a collection of objects according to certain characteristics
2. The objects in a set are known as elements.
3. Sets are usually denoted by capital letters and notation used for sets is braces, {   }.

Example:
A = {1, 3, 5, 7, 9}
 
4. In set notation, the symbol ∈ means ‘is an element of’ or ‘belongs to’ and ∉ means ‘is not an element of’ or ‘does not belong to’.
 
Example 1:
Given that P= {factors of 15} and Q = {positive perfect squares less than 28}. By using the symbol ∈ or ∉ , 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) 5 ∈ P ← 5 is an element of set P (b) 20 ∉ P ← 20 is not an element of set P (c) 2 5 ∈ Q ← 2 5 is an element of set Q (d) 8 ∉ Q ← 8 is not an element of set Q



(A) Represent sets by using Venn diagram
5. A set can be represented by a Venn diagram using closed geometry shapes such as circles, rectangles, triangles and etc.

6. A dot to the left of an object in a Venn diagram indicates that the object is an element of the set.

7. When a Venn diagram represents the number of elements in a set, no dot is 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) = 4
n(B) = 5



(B) Determine whether a set is an empty set
8. A set with no elements is called an empty set or null set. The symbol φ or empty braces, {  }, denotes empty set.
For example, if set A is an empty set, then A = {  } or A = φ and
n (A) = 0.
 
9. If B = {0} or {φ} does not denote that B is an empty set. B = {0} means that there is an element ‘0’ in set B.
B = {φ} means that there is an element ‘φ’ in set B.