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)  $B \subset C \subset A$

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)  $C \subset B \subset A$

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

(iii)  $C \subset D \subset A$

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

(iv)  $D \subset C \subset A$

Note: All rhombuses are not squares.

(v)  $A \supseteq B \supseteq C$

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)  $A \subseteq B \subseteq C$

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)  $B \subset C \subset A$

Note: All isosceles triangles are not equilateral triangles.

(ii)  $C \subset B \subset A$

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)  $1 \subset A$

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

(ii)  $\{ 1 \} \in A$

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

(iii)  $1 \in A$

Note: $1$ is an element of set $A$.

(iv)  $\phi \in A$

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

(v)  $\phi \subset A$

Note: Null set is a subset of set $A$.

(vi)  ${1} \notin A$

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)  $a \subset \{ a, b, c \}$

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

(ii)  $\{a \} \in \{a, b, c \}$

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

(iii)  $\{a \} \subset \{a, b, c \}$

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

(iv)  $\phi \in \{a, b, c \}$

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

(v)  $\phi \subset \{a, b, c \}$

(vi)  $\{ \phi \} \subset \{a, b, c \}$

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

(vii)  $a \in \{ \{a \}, b \}$

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

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

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

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

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) $\phi = \{0 \}$

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

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

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

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

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

(iv) $\phi \in { \phi}$

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

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

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

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

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 $B \subseteq A$

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

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

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

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

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.

$\xi = {1, 2, 3, 4, ,5, ..., 49}$. 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) $A \subseteq B$

(ii) $A = B$

(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 $n(C) = 9$