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Page No 66:
Question O1:
What is a set?
Answer:
A set is a collection of well defined objects or numbers. Every element of a set is called the set element.
Page No 66:
Question W1.1:
Write which is a set and which is not a set in the groups given below.
Counting numbers
Answer:
Counting numbers is a set. It is because, counting numbers are well defined objects, for instance 1, 2, 3, … .
Page No 66:
Question W1.2:
Write which is a set and which is not a set in the groups given below.
Good cricket players of your school
Answer:
Goodness of a cricket players may vary from person to person.”
So, good cricket players are not well defined.
Thus, good cricket players of our school are not a set.
Page No 66:
Question W1.3:
Write which is a set and which is not a set in the groups given below.
All negative whole numbers
Answer:
All negative whole numbers is a set because negative whole numbers are well defined objects, for instance −1, −2, −3, ….
Page No 66:
Question W1.4:
Write which is a set and which is not a set in the groups given below.
All intelligent students of mathematics
Answer:
All intelligent students of mathematics are not a set because intelligence of a student varies from person to person. So, intelligent students of mathematics are not well defined.
Page No 66:
Question O2:
Which are the two methods of writing sets?
Answer:
The two methods of writing sets are roster method and rule method.
Page No 66:
Question W1.5:
Write which is a set and which is not a set in the groups given below.
Digits used in base five system.
Answer:
Collection of digits used in base five systems is a set because digits used in base five systems are well defined objects, for instance 0, 1, 2, 3 and 4.
Page No 66:
Question W2:
Write suitable answers in the blanks.
Rule Method |
Roster Method |
||
1. |
---------------- |
1. |
O = {1, 3, 5, 7, 9} |
2. |
{months in a year} |
2. |
---------------- |
3. |
---------------- |
3. |
V = {a, e, i, o, u} |
4. |
{numbers on the face of a clock} |
4. |
---------------- |
Answer:
Rule method |
Roster method |
1. (odd numbers from 1 to 10) |
1. O = {1, 3, 5, 7, 9} |
2. (months in a year) |
2. M= {January, February, March, April, May, June, July, August, September, October, November, December } |
3. (vowels of English alphabet) |
3. V = {a, e, i, o, u} |
4. (numbers on the face of a clock) |
4. N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} |
Page No 66:
Question O3:
What are equivalent sets?
Answer:
The sets which have same number of elements are called equivalent sets.
Page No 66:
Question W3.1:
Write the following sets in rule method.
K = {5, 10, 15, 20, 25 ………….}
Answer:
K = {multiples of 5}
Page No 66:
Question W3.2:
Write the following sets in rule method.
L = {summer season, rainy season, winter season}
Answer:
L = {Seasons of the year}
Page No 67:
Question W5.5:
State whether the following are ‘True’ or ‘False’
A set of population of the world is an example for a finite set.
Answer:
True
Reason: Population of the world can be counted.
Page No 69:
Question O1:
Mention the name of a Mathematician who showed the method of representing sets through diagrams.
Answer:
British mathematician, John Venn in the 19^{th} century showed the method of representing sets through diagrams. The diagram used in sets is called Venn Diagrams.
Page No 70:
Question W1:
If
Solve the following problems. Draw Venn diagrams for each of them.
Answer:
A = {2, 3, 4, 5, 6}
B = {2, 4, 6, 8, 10}
C = {3, 6, 7, 9}
(1) A ∩ B = {2, 3, 4, 5, 6} ∩ {2, 4, 6, 8, 10}
= {2, 4, 6}
And the Venn diagram is:
(2) B ∩ C = {2, 4, 6, 8, 10} ∩ {3, 6, 7, 9}
= {6}
And the Venn diagram is:
(3) A ∩ C = {2, 3, 4, 5, 6} ∩ {3, 6, 7, 9}
= {3, 6}
And the Venn diagram is:
(4) (A ∩ B) ∩ C = {2, 4, 6} ∩ {3, 6, 7, 9} (From (1) A ∩ B = {2, 4, 6})
= {6}
And the Venn diagram is:
(5) A ∩ (B ∩ C) = {2, 3, 4, 5, 6} ∩ {6} (From (2) B ∩ C = {6})
= {6}
And the Venn diagram is:
Page No 70:
Question W2:
If
Solve the following problems and draw Venn diagrams for each of them.
Answer:
P = {a, b, c, d, e}
Q = {a, e, i, o, u}
R = {a, c, e, g, i}
(1) P ∩ Q = {a, b, c, d, e} ∩ {a, e, i, o, u}
= {a, e}
And the Venn diagram is:
(2) Q ∩ R = {a, e, i, o, u} ∩ {a, c, e, g, i}
= {a, e, i}
And the Venn diagram is:
(3) P ∩ R = {a, b, c, d, e} ∩ {a, c, e, g, i}
= {a, c, e}
And the Venn diagram is:
(4) (P ∩ Q) ∩ R = {a, e} ∩ {a, c, e, g, i} (From (1) P ∩ Q = {a, e})
= {a, e}
And the Venn diagram is:
(5) P ∩ (Q ∩ R) = {a, b, c, d, e} ∩ {a, e, i} (From (2) Q ∩ R= {a, e, i})
= {a, e}
And the Venn diagram is:
Page No 70:
Question W3:
If
Solve the following problems and draw Venn diagrams for each of them.
Answer:
X = {2, 3, 4, 5, 6}
Y = {1, 7, 8}
Z = {1, 3, 5, 7, 9}
(1) X ∪ Y = {2, 3, 4, 5, 6} ∪ {1, 7, 8}
= {1, 2, 3, 4, 5, 6, 7, 8}
And the Venn diagram is:
(2) Y ∪ Z = {1, 7, 8} ∪ {1, 3, 5, 7, 9}
= {1, 3, 5, 7, 8, 9}
And the Venn diagram is:
(3) X ∪ Z = {2, 3, 4, 5, 6} ∪ {1, 3, 5, 7, 9}
= {1, 2, 3, 4, 5, 6, 7, 9}
And the Venn diagram is:
(4) (X ∪ Y) ∪ Z = {1, 2, 3, 4, 5, 6, 7, 8} ∪ {1, 3, 5, 7, 9}
(From (1) X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8} )
= {1, 2, 3, 4, 5, 6, 7, 8, 9}
And the Venn diagram is:
(5) X ∪ (Y ∪ Z) = {2, 3, 4, 5, 6} ∪ {1, 3, 5, 7, 8, 9}
(From (2) Y ∪ Z = {1, 3, 5, 7, 8, 9})
= {1, 2, 3, 4, 5, 6, 7, 8, 9}
And the Venn diagram is:
Page No 70:
Question W4:
If
Solve the following problems and draw Venn diagrams for each of them.
Answer:
M = {g, b, c}
N = {a, e, i, o, u}
P = {v, w, x, y, z}
(1) M ∪ N = {g, b, c} ∪ {a, e, i, o, u}
= {a, b, c, e, g, i, o, u}
And the Venn diagram is:
(2) N ∪ P = {a, e, i, o, u} ∪ {v, w, x, y, z}
= { a, e, i, o, u, v, w, x, y, z}
And the Venn diagram is:
(3) M ∪ P = {g, b, c} ∪ {v, w, x, y, z}
= { g, b, c, v, w, x, y, z}
And the Venn diagram is:
(4) (M ∪ N) ∪ P = {g, b, c, a, e, i, o, u} ∪ {v, w, x, y, z}
(From (1) M ∪ N = { g, b, c, a, e, i, o, u })
= {a, b, c, e, g, i, o, u, v, w, x, y, z}
And the Venn diagram is:
(5) M ∪ (N ∪ P) = {g, b, c} ∪ { a, e, i, o, u, v, w, x, y, z}
(From (2) N ∪ P = { a, e, i, o, u, v, w, x, y, z}
= { g, b, c, a, e, i, o, u, v, w, x, y, z}
And the Venn diagram is:
Page No 70:
Question W5:
In a class of 30 students, 20 of them are interested in mathematics, 25 students are interested in social science. Find out the number of students who are interested in both and show the same through a Venn diagram.
Answer:
Total number of students in a class = 30
Total number of students interested in mathematics and social science = 20 + 25 = 45
Number of students interested in both mathematics and science = 45 − 30 = 15
And the Venn diagram is:
Page No 70:
Question W6.1:
In a group of students, D = {a, b, c, d, e, f} is the set of students participating in drama and M = {d, g, a, h} is the set of students participating in music.
Draw Venn diagrams for the given data:
The set of students participating only in drama
Answer:
D = {a, b, c, d, e, f}
M = {d, g, a, h}
Set of students participating only in drama = D - (D ∩ M)
= {a, b, c, d, e, f}-({a, b, c, d, e, f} ∩ {d, g, a, h})
= {a, b, c, d, e, f} - {a, d}
= {b, c, e, f}
And the Venn diagram is:
Page No 70:
Question W6.2:
In a group of students, D = {a, b, c, d, e, f} is the set of students participating in drama and M = {d, g, a, h} is the set of students participating in music.
Draw Venn diagrams for the given data:
The set of students participating only in music
Answer:
D = {a, b, c, d, e, f}
M = {d, g, a, h}
Set of students participating only in music = M- (M ∩ D)
= {d, g, a, h} - ({d, g, a, h} ∩ {a, b, c, d, e, f})
= {d, g, a, h} - {a, d}
= {g, h}
And the Venn diagram is:
Page No 70:
Question W6.3:
In a group of students, D = {a, b, c, d, e, f} is the set of students participating in drama and M = {d, g, a, h} is the set of students participating in music.
Draw Venn diagrams for the given data:
The set of students participating both in drama and music
Answer:
D = {a, b, c, d, e, f}
M = {d, g, a, h}
Set of students participating both in drama and music = D ∩ M
= {a, b, c, d, e, f}∩{d, g, a, h}
= {a, d}
And the Venn diagram is:
Page No 70:
Question W6.4:
In a group of students, D = {a, b, c, d, e, f} is the set of students participating in drama and M = {d, g, a, h} is the set of students participating in music.
Draw Venn diagrams for the given data:
Total number of students participating either in drama or in music or both
Answer:
D = {a, b, c, d, e, f}
M = {d, g, a, h}
Total number of students participating either in drama or in music, or both = D ∪ M
= {a, b, c, d, e, f}∪{d, g, a, h}
= {a, b, c, d, e, f, g, h}
And the Venn diagram is:
Page No 72:
Question O1:
Mention the laws which are applicable in the following situations.
Answer:
(a) The property used in is commutative property.
(b) The property used in is associative property.
Page No 72:
Question W1.1:
If
K ∪ L = L ∪ K
Answer:
K = {11, 12, 13}
L = {12, 13, 14}
M= {13, 14, 15}
K ∪ L = {11, 12, 13} ∪ {12, 13, 14}
= {11, 12, 13, 14}
L ∪ K = {12, 13, 14} ∪ {11, 12, 13}
= {11, 12, 13, 14}
∴ K ∪ L = L ∪ K
Page No 72:
Question W1.2:
If
K ∪ M = M ∪ K
Answer:
K = {11, 12, 13}
L = {12, 13, 14}
M= {13, 14, 15}
K ∪ M = {11, 12, 13} ∪ {13, 14, 15}
= {11, 12, 13, 14, 15}
M ∪ K = {13, 14, 15} ∪ {11, 12, 13}
= {11, 12, 13, 14, 15}
∴ K ∪ M = M ∪ K
Page No 72:
Question W1.3:
If
M ∪ L = L ∪ M
Answer:
K = {11, 12, 13}
L = {12, 13, 14}
M= {13, 14, 15}
M∪ L = {13, 14, 15} ∪ {12, 13, 14}
= {12, 13, 14, 15}
L ∪ M ={12, 13, 14} ∪ {13, 14, 15}
= {12, 13, 14, 15}
∴ M∪ L = L ∪ M
Page No 72:
Question W1.4:
If
K ∩ L = L ∩ K
Answer:
K = {11, 12, 13}
L = {12, 13, 14}
M= {13, 14, 15}
K ∩ L = {11, 12, 13} ∩ {12, 13, 14}
= {12, 13}
L ∩ K = {12, 13, 14} ∩ {11, 12, 13}
= {12, 13}
∴ K ∩ L = L ∩ K
Page No 72:
Question W1.5:
If
K ∩ M = M ∩ K
Answer:
K = {11, 12, 13}
L = {12, 13, 14}
M= {13, 14, 15}
K ∩ M = {11, 12, 13} ∩ {13, 14, 15}
= {13}
M ∩ K = {13, 14, 15}∩ {11, 12, 13}
= {13}
∴ K ∩ M = M ∩ K
Page No 72:
Question W1.6:
If
L ∩ M = M ∩ L
Answer:
K = {11, 12, 13}
L = {12, 13, 14}
M= {13, 14, 15}
L ∩ M = {12, 13, 14} ∩ {13, 14, 15}
= {13, 14}
M ∩ L = {13, 14, 15} ∩ {12, 13, 14}
= {13, 14}
∴ L ∩ M = M ∩ L
Page No 72:
Question W2.1:
If
Prove that (X ∩ Y) ∩ Z = X ∩ (Y ∩ Z)
Answer:
X = {10, 11, 12}
Y = {12, 13, 14}
Z = {14, 15, 16}
X ∩ Y = {10, 11, 12} ∩ {12, 13, 14}
= {12}
(X ∩ Y) ∩ Z = {12} ∩ {14, 15, 16}
= { }
= Φ
Y ∩ Z = {12, 13, 14} ∩ {14, 15, 16}
= {14}
X ∩ (Y ∩Z) = {10, 11, 12} ∩ {14}
= { }
= Φ
∴ (X ∩ Y) ∩ Z = X ∩ (Y ∩Z)
Page No 72:
Question W2.2:
If
Show that (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z)
Answer:
X = {10, 11, 12}
Y = {12, 13, 14}
Z = {14, 15, 16}
X ∪ Y = {10, 11, 12} ∪ {12, 13, 14}
= {10, 11, 12, 13, 14}
(X ∪ Y) ∪ Z = {10, 11, 12, 13, 14} ∪ {14, 15, 16}
= {10, 11, 12, 13, 14, 15, 16}
Y ∪ Z = {12, 13, 14} ∪ {14, 15, 16}
= {12, 13, 14, 15, 16}
X ∪ (Y ∪ Z) = {10, 11, 12} ∪ {12, 13, 14, 15, 16}
= {10, 11, 12, 13, 14, 15, 16}
∴(X ∪ Y) ∪ Z = X ∪ (Y ∪ Z)
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