Question 1. Write down the following statements, using set notation:

(i)  Set A is a proper subset of set B

Answer: A \subset B

Note: Let A be any set and B a non-empty set. Then, A is called a proper subset of B is all elements of A exists in the set B but B has at least one element that is not in set A .

(ii)  Set C is a superset of set D

Answer:  C \supseteq D

Note: If A is a subset of set B , then B is called the super set of A .

(iii)  Set B contains set A

Answer: B \supseteq A

Note: If B contains set A , that means that B is a super set of A .

(iv)  Neither A is a subset of B , nor B is a subset of A

Answer: A \not\subset B and B \not\subset A

Note: If there exists even a single element in set A that does not exist in the set B , then A is not a subset of B . Similarly, if there is any element in B that does not exist in set A , then B is not a subset of A .

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Question 2. Let A = \{ all \ quadrilaterals \} , B = \{ all \ rectangles \} , C = \{ all \ squares \} and D = \{ all \ rhombuses \} in a plane. State, giving reasons, whether the following statements are true or false.

(i)  B \subset C \subset A

Answer: False

Note: All rectangles are not squares and hence B is not a proper subset of C. All squares are quadrilaterals and hence C is a proper subset of A.

(ii)  C \subset B \subset A

Answer: True

Note: All squares are also rectangles, and all rectangles are quadrilaterals.

(iii)  C \subset D \subset A

Answer: True

Note: All squares are rhombuses and all rhombuses are quadrilaterals.

(iv)  D \subset C \subset A

Answer: False

Note: All rhombuses are not squares.

(v)  A \supseteq B \supseteq C

Answer: True

Note: A is a super set of B and B is a super set of C. This is because, all quadrilaterals will contain all rectangles and all rectangles would contain all squares.

(vi)  A \subseteq B \subseteq C

Answer: False

Note: All rectangles need not contain all quadrilaterals. And all squares need not contain all rectangles.

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Question 3. Let A = \{ all \ triangles \} , B = \{ all \ isosceles \ triangles \} and C = \{ all \ equilateral \ triangles \} . State, giving reasons, whether the following statements are true or false

(i)  B \subset C \subset A

Answer:  False

Note: All isosceles triangles are not equilateral triangles.

(ii)  C \subset B \subset A

Answer: True

Note: All equilateral triangles are isosceles triangles and all isosceles triangles are triangles

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Question 4. Let A = \{ 1, 2 \} . State which of the following statements are true:

(i)  1 \subset A

Answer: False

Note: 1 is an element of set A . 1 is not a set.

(ii)  \{ 1 \} \in A

Answer: False

Note: \{ 1 \} is a set. It does not belong to A . The element 1 belongs to A .

(iii)  1 \in A

Answer: True

Note: 1 is an element of set A .

(iv)  \phi \in A

Answer: False

Note: \phi does not belong to set A . \phi means a null or empty set.

(v)  \phi \subset A

Answer: True

Note: Null set is a subset of set A .

(vi)  {1} \notin A

Answer: False

Note: \{ 1 \} is a subset. Element belong or not belong to set A .

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Question 5. Which of the following statements are correct?

(i)  a \subset \{ a, b, c \}

Answer: False

Note: a is an element of the set \{a, b, c \} . Hence it should be a \in \{a, b, c \}

(ii)  \{a \} \in \{a, b, c \}

Answer: False

Note: \{a \} is subset of the set \{a, b, c \} . Hence it should be a \subset  \{a, b, c \}

(iii)  \{a \} \subset \{a, b, c \}

Answer: Correct

Note: \{a \} is subset of the set \{a, b, c \} . Hence it should be a \subset  \{a, b, c \}

(iv)  \phi \in \{a, b, c \}

Answer: False

Note: \phi  is not an element of \{a, b, c \} . It should be \phi \subset \{a, b, c \}

(v)  \phi \subset \{a, b, c \}

Answer: Correct

(vi)  \{ \phi \} \subset \{a, b, c \}

Answer: False

Note: \{a, b, c \} does not contain \{ \phi \}

(vii)  a \in \{ \{a \}, b \}

Answer: False

Note: a does not belong to \{ \{a \}, b \} , but \{a \} belongs to \{ \{a \}, b \}

(viii)  \{a \} \subseteq \{ \{a \}, b \}

Answer: False

Note: \{a \} is an element of the set \{ \{a \}, b \} and not a set.

(ix)  \{a, b \} \in \{ \{a, b \}, c \}

Answer: Correct

Note: \{a, b \} belong to \{ \{a, b \}, c \} as \{a, b \} is an element of the set \{ \{a, b \}, c \}

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Question 6. Which of the following statements are true?

(i) \phi = \{0 \}

Answer: False

Note: The set contains 0 as an element. It is not a null set.

(ii) \phi = \{ \phi \}

Answer: False

Note: The set \phi contain as the element. It is not a null set.

(iii) \phi \in \{ 0 \}

Answer: False

Note: \phi does not belong to the set \{ 0 \}

(iv) \phi \in { \phi}

Answer: True

Note: \phi is an element in the set \{ \phi \}

(v) \phi \in \{ \phi, \{0 \} \}

Answer: True

Note:\phi is an element of the set \{ \phi, \{0 \} \}

(vi) \phi \subset \{0 \}

Answer: False

Note: \phi \subseteq \{0 \}

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Question 7. Which of the following statement are true?

(i) For any two sets A and B , either A \subseteq B or B \subseteq A

Answer: False

Note: For A to be a subset of B , all elements of A should be in B . This need not be true as A can have elements that are not in B and B can have elements that are not in A . Hence it is not necessary that for any two sets A and B , either A \subseteq B or B \subseteq A 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 A = \{ 1, 2, 3, 4, 5, ... \} . Then \{ 1 \} is a subset of A . Though A is infinite, the subset \{ 1 \} 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 A has n elements, then P(A) has 2^n subsets

Answer: True

Note: The set of all possible subsets of a set A is called the power set of A , and is denoted by P(A) . If A $ contains n elements, then P(A) contains 2^n  subsets.

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Question 8. Let A be the set of letters in the word ‘seed’. Find

(i) A

Answer: A = \{ s, e, d \}

(ii) n(A)

Answer: 3

(iii) Number of subset of A

Answer: 2^3=8

Note: The set of all possible subsets of a set A is called the power set of A , and is denoted by P(A) . If A contains n elements, then P(A) contains 2^n subsets.

(iv) Number of proper subsets of A

Answer: ( 2^3 - 1 ) = 7

Note: A set containing n elements has (2^n - 1) proper subsets.

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Question 9. Find all possible subsets of each of the following sets:

(i ) A = \{4, 9 \}

Answer: \phi, \{4 \}, \{9 \}, \{4, 9 \}

Note: Number of elements = 2 . Total subsets = 2^2 = 4

(ii) B = \{2, 3, 8 \}

Answer: \phi, \{2 \}, \{3 \}, \{8 \}, \{2, 3 \}, \{2, 8 \}, \{3, 8 \}, \{2, 3, 8 \}

Note: Number of elements = 3 . Total subsets = 2^3 = 8

(iii) C = \{0, 1, 2 \}

Answer: \phi, \{0 \}, \{1 \}, \{2 \}, \{0, 1 \}, \{0, 2 \}, \{1, 2 \}, \{0, 1, 2 \}

Note: Number of elements = 3 . Total subsets = 2^3 = 8

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Question 10. Find the power set of each of the following:

(i) A = \{0, 5 \}

Answer: P(A) = \{ \phi, \{ 0 \} , \{ 5 \} , \{ 0, 5 \}  \}

Note: P(A) = Set of all possible subsets of set A

(ii) B = \{7, 9 \}

Answer: P(A) = \{ \phi, \{ 7 \} , \{ 9 \} , \{ 7, 9 \}  \}

(iii) C = \{2, 4, 6 \}

Answer: P(A) = \{ \phi, \{ 2 \} , \{ 4 \} , \{ 6 \}, \{ 2, 4 \} , \{ 2, 6 \} , \{ 4, 6 \}, \{ 2, 4, 6 \}   \}

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Question 11. Let A = \{1 , \{ 2 \} \} . Find the power of set A

Answer: P(A) = \{ \phi, \{1\}, \{ \{ 2 \} \} , \{ 1, \{ 2 \} \}

Note: consider \{ 2 \} as an element.

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Question 12. Let x = \{ x : x \in N, x < 50 \} , A = \{ x : x^2 \in \xi \} B - \{ x: x = n^2, n \in N \} and C = \{ x : x \ is \ a \ factor \ of \ 36 \} . List all the elements of set A, B , and C . Also state whether each of the following statement is true or false.

Answer:

\xi = {1, 2, 3, 4, ,5, ..., 49} . Basically x is values from 1 to 49 . Therefore:  

A = {1, 2, 3, 4, 5, 6, 7} .

B = {1, 4, 9, 16, 25, 36, 49}

C = {1, 2, 3, 4, 6, 9, 12, 18, 36} . All factors of 36 but within 1 \leq x \leq 49

(i) A \subseteq B

Answer: False

(ii) A = B

Answer: False

(iii) A \leftrightarrow B

Answer: True since n(A) = 7   while n(B) = 7

(iv) B \leftrightarrow C

Answer: False since n(B) = 7   while n(C) = 9

(v) n(A) < n(C)

Answer: True since n(A) = 7   while n(C) = 9