Question 1. Write down the following statements, using set notation:
(i) Set is a proper subset of set
Answer:
Note: Let be any set and
a non-empty set. Then,
is called a proper subset of
is all elements of
exists in the set
but
has at least one element that is not in set
.
(ii) Set is a superset of set
Answer:
Note: If is a subset of set
, then
is called the super set of
.
(iii) Set contains set
Answer:
Note: If contains set
, that means that
is a super set of
.
(iv) Neither is a subset of
, nor
is a subset of
Answer: and
Note: If there exists even a single element in set that does not exist in the set
, then
is not a subset of
. Similarly, if there is any element in
that does not exist in set
, then
is not a subset of
.
Question 2. Let ,
,
and
in a plane. State, giving reasons, whether the following statements are true or false.
(i)
Answer: False
Note: All rectangles are not squares and hence is not a proper subset of
. All squares are quadrilaterals and hence
is a proper subset of
.
(ii)
Answer: True
Note: All squares are also rectangles, and all rectangles are quadrilaterals.
(iii)
Answer: True
Note: All squares are rhombuses and all rhombuses are quadrilaterals.
(iv)
Answer: False
Note: All rhombuses are not squares.
(v)
Answer: True
Note: A is a super set of and
is a super set of
. This is because, all quadrilaterals will contain all rectangles and all rectangles would contain all squares.
(vi)
Answer: False
Note: All rectangles need not contain all quadrilaterals. And all squares need not contain all rectangles.
Question 3. Let ,
and
. State, giving reasons, whether the following statements are true or false
(i)
Answer: False
Note: All isosceles triangles are not equilateral triangles.
(ii)
Answer: True
Note: All equilateral triangles are isosceles triangles and all isosceles triangles are triangles
Question 4. Let . State which of the following statements are true:
(i)
Answer: False
Note: is an element of set
.
is not a set.
(ii)
Answer: False
Note: is a set. It does not belong to
. The element
belongs to
.
(iii)
Answer: True
Note: is an element of set
.
(iv)
Answer: False
Note: does not belong to set
.
means a null or empty set.
(v)
Answer: True
Note: Null set is a subset of set .
(vi)
Answer: False
Note: is a subset. Element belong or not belong to set
.
Question 5. Which of the following statements are correct?
(i)
Answer: False
Note: is an element of the set
. Hence it should be
(ii)
Answer: False
Note: is subset of the set
. Hence it should be
(iii)
Answer: Correct
Note: is subset of the set
. Hence it should be
(iv)
Answer: False
Note: is not an element of
. It should be
(v)
Answer: Correct
(vi)
Answer: False
Note: does not contain
(vii)
Answer: False
Note: a does not belong to , but
belongs to
(viii)
Answer: False
Note: is an element of the set
and not a set.
(ix)
Answer: Correct
Note: belong to
as
is an element of the set
Question 6. Which of the following statements are true?
(i)
Answer: False
Note: The set contains as an element. It is not a null set.
(ii)
Answer: False
Note: The set contain as the element. It is not a null set.
(iii)
Answer: False
Note: does not belong to the set
(iv)
Answer: True
Note: is an element in the set
(v)
Answer: True
Note: is an element of the set
(vi)
Answer: False
Note:
Question 7. Which of the following statement are true?
(i) For any two sets and
, either
or
Answer: False
Note: For to be a subset of
, all elements of
should be in
. This need not be true as
can have elements that are not in
and
can have elements that are not in
. Hence it is not necessary that for any two sets
and
, either
or
is always true.
(ii) Every subset of a finite set is a finite set
Answer: True
Note: Finite set is a set where the counting process of the elements comes to an end. Basically, the set contains finite elements that can be counted. So, if the set has finite elements, then all the subsets will also be finite.
(iii) Every subset of an infinite set is infinite
Answer: False
Note: Let . Then
is a subset of
. Though
is infinite, the subset
is finite.
(iv) Every set has a proper subset
Answer: False
Note: No. The null set cannot have a proper subset. For any other set, the null set will be a proper subset. There will also be other proper subsets.
(v) If has
elements, then
has
subsets
Answer: True
Note: The set of all possible subsets of a set is called the power set of
, and is denoted by
. If
contains
elements, then
contains
subsets.
Question 8. Let A be the set of letters in the word ‘seed’. Find
(i)
Answer:
(ii)
Answer:
(iii) Number of subset of
Answer:
Note: The set of all possible subsets of a set is called the power set of
, and is denoted by
. If
contains
elements, then
contains
subsets.
(iv) Number of proper subsets of
Answer:
Note: A set containing elements has
proper subsets.
Question 9. Find all possible subsets of each of the following sets:
(i )
Answer:
Note: Number of elements . Total subsets
(ii)
Answer:
Note: Number of elements . Total subsets
(iii)
Answer:
Note: Number of elements . Total subsets
Question 10. Find the power set of each of the following:
(i)
Answer:
Note: Set of all possible subsets of set
(ii)
Answer:
(iii)
Answer:
Question 11. Let . Find the power of set
Answer:
Note: consider as an element.
Question 12. Let ,
and
. List all the elements of set
, and
. Also state whether each of the following statement is true or false.
Answer:
. Basically
is values from
to
. Therefore:
.
. All factors of
but within
(i)
Answer: False
(ii)
Answer: False
(iii)
Answer: True since while
(iv)
Answer: False since while
(v)
Answer: True since while