Question 1: Which of the following collections of objects are sets?
(i) All the months in a year
Answer: YES. It is a well-defined collection of distinct objects. There are months in a year. So, it is a definite set of elements.
(ii) All the rivers flowing in UP
Answer: YES. It is a well-defined collection of distinct objects.
(iii) All the planets in our solar system
Answer: YES. It is a well-defined collection of distinct objects.
(iv) All the interesting dramas written by Premchand
Answer: NO. It is not a well-defined collection of objects. Had it been “All dramas written by Premchand” then it could have been a set as that would be well defined.
(v) All the short boys in your class
Answer: NO. It is not a well-defined collection of objects. Had it been “All boys shorter than , then it would have become well defined. We need a benchmark to have a well-defined list of objects.
(vi) All the letters of the English Alphabets which precedes K
Answer: YES. It is a well-defined collection of distinct objects.
(vii) All the pet dogs in Nagpur
Answer: YES. It is a well-defined collection of distinct objects. Technically, all pets in Nagpur should be registered with a Government department. This might be a big list but still a well-defined list of objects.
(viii) All the dishonest shop owners in Noida
Answer: No. Not well defined.
(ix) All the students in your school with age exceeding 15 years
Answer: YES. It is a well-defined collection of distinct objects. School has a list of students that study there. The set would include all whose age is more than years.
(x) All the girls of Gita’s class who are taller than Gita
Answer: YES. It is a well-defined collection of distinct objects. We know all the girls studying in the class, we know Gita’s height and hence the set would contain distinct girls taller than Gita.
Question 2: Rewrite the following statements using the set notations:
(i) is an element of
Answer:
(ii) does not belong to set
Answer:
(iii) and
are members of set
Answer:
(iv) and
are equivalent sets
Answer:
(v) Cardinal number of set is
Answer:
Note: The number of distinct elements contained in a finite set is called the cardinal number of
and is denoted by
.
For example, if , then
(vi) is an empty set and
is a non-empty set
Answer: and
Note: A set consisting of no elements is called an empty set or a null set or a void set. It is denoted by (called phai). We write
(vii) is a whole number, but
is not a natural number
Answer: but
Note: Whole numbers : The numbers
.
Natural numbers : The counting numbers
, are called natural numbers
Question 3: Describe the following sets in roster form
(i)
Answer:
(ii)
Answer:
Note: You can calculate the factors using a tree method.
(iii)
Answer:
Note: and
. This means
is
, and
. Now calculate
based on the given formula. Example: When
.
(iv)
Answer:
Note: and
. This means
is
. Now substitute the value of
to calculate
.
(v)
Answer:
Note: and
, which means
is
. Now substitute the value of
in the formula given for
.
(vi)
Answer:
Note: If the first digit is , then the second digit is
. So now try the values of
as
cannot be
as it is a two digit number. Also by same logic,
cannot be greater than
either. Hence
can only be
and
only.
(vii)
Answer:
Note: means that
. Of these numbers
needs to be divisible both by
and
.
(viii)
Answer:
Note: and
which means that
is
. Now substitute the values.
(ix)
Answer:
Note: In careless, the alphabet and
are repeated. We only need to take into account distinct elements only.
Question 4: Describe the following sets in set builder form
(i)
Answer:
(ii)
Answer:
(iii)
Answer:
(iv)
Answer: or
(v)
(vi)
Answer:
Note: If you notice, the numbers are square of .
(vii)
Answer:
Note: Integers (Z): Positive and negative counting numbers, as well as zero:
(viii) which is
which means
can be
or
(ix)
Answer:
(x)
Answer:
(xi)
Answer:
(xii)
Answer:
(xiii)
Answer:
(xiv)
Answer:
Question 5: Separate finite and infinite sets from the following:
(i) Set of leaves on a tree
Answer: Finite. This is because in this case, the process of counting the leaves would surely come to an end.
(ii) Set of all counting numbers
Answer: Infinite. This is because there is no end to the numbers.
(iii)
Answer: Infinite. This is because there is no end to the numbers since .
(iv)
Answer: Finite. is whole numbers which are all natural numbers including
. Since
is less than
, there are finite numbers to be counted.
(v)
Answer: Infinite. is an integer. Integers can be negative numbers too. So in this case, though
is limited to less than
on the positive scale, it can go to infinity on the negative scale.
(vi) Set of all triangles in a plane
Answer: Infinite. This is because there can be uncountable number of triangles in a plane.
(vii) Set of all points on a circumference of a circle
Answer: Infinite. This is because there can be uncountable number of points on a circumference of a circle.
(viii)
Answer: Infinite. This is because the set is not ending. It will keep having 1, 2, 3 repeated again and again
(ix)
Answer: Infinite.
Note: Rational numbers (Q): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.
Question 6: Which of the following are empty sets?
(i)
Answer: Empty Set
Note: Natural Number . Since
is given, it means,
. But
does not belong to
and hence the set it empty or null.
(ii)
Answer: Empty Set
Note: Natural Number . Since
is given, it means,
. But
does not belong to
and hence the set it empty or null.
(iii)
Answer: Not an Empty Set
Note: Whole Number . Since
is given, it means,
. Therefore
can be
and hence it is not an Empty Set.
(iv)
Answer: Not an Empty Set
Note: Natural Number . Since
, it means that
can be
. Hence
is not an Empty Set.
(v)
Answer: Empty Set
Note: Natural Number .
means that
is
which is not possible.
(vi)
Answer: Empty Set
Note: If , this means
can be
. All these numbers are divisible by a number. E.g.
is divisible by
by
by
and so on.
(vii)
Answer: Not an Empty Set.
Note: is an even prime number.
Question 7: Which of the following are pairs of equivalent sets?
Note: Two finite sets and
are said to be equivalent, if
, that is they have the same number of elements. An equivalent set is simply a set with an equal number of elements. The sets do not have to have the same exact elements, just the same number of elements.
(i) and
Answer: No
Note: and
Therefore
while
(ii) and
Answer: No
Note: and
Therefore
while
(iii) and
Answer: Yes
Note: and
Therefore
while
Hence
and
are equivalent sets.
(iv) and
Answer: Yes
Note: and
Therefore
and
Hence they are equivalent sets
Question 8: State whether the following statements are true or false.
(i) is not a set
Answer: False. It is a set. It contains elements.
(ii)
Answer: True. Both sets contain the same elements just in different order. The order of elements does not matter.
(iii) is a singleton set
Answer: True. The only element in the set would be . Hence it is a singleton set.
Note: a singleton, also known as a unit set, is a set with exactly one element. For example, the set is a singleton. The term is also used for a 1-tuple (a sequence with one element).
(iv)
Answer: True. The set is a null set as there is no element.
(v)
Answer: True. but
is a natural number. Hence a null set.
(vi)
Answer: False. There is a valid element in the set which is .
(vii) If then
Answer: False.
(viii) If and
, then
Answer: True. . Therefore
.
Therefore
. Hence
(ix) If , then
Answer: False. Just because the number of elements in the sets and
are equal, does not mean that the sets are also equivalent.
(x) If is a set of all consonants in English Alphabets, then
Answer: True
Note: The word consonant is also used to refer to a letter of an alphabet that denotes a consonant sound. The consonant letters in the English alphabet are
, and usually
and
.
(xi)
Answer: False. Prime factors of are
, and