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