Question 1: Which of the following statements are true? Give reason to support your answer.

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

(ii) Every subset of an infinite set is infinite

(iii) Every subset of a finite set is finite

(vi) Every set has a proper subset

(v) $\{a, b, a, b, a, b,...)$ is an infinite set.

(vi) $\{a, b, c\}$ and $\{1,2,3\}$ are equivalent sets

(vii) A set call have infinitely many subsets.

(i) For any two sets $A$ and $B$ either $A \subseteq B$ or $B \subseteq A$: FALSE – This is not always necessary. For example $A = \{x, y, z \}$ and $B = \{a, b, c \}$.  In that case neither $A \subseteq B$ or $B \subseteq A$

(ii) Every subset of an infinite set is infinite: FALSE$A = \{ 2 \}$ is a finite subset of $N$ (which is infinite) or we can say $A = \{ -1, 0, 1 \}$ is a finite subset of $Z$.

(iii) Every subset of a finite set is finite: TRUE

(iv) Every set has a proper subset: FALSE$\phi$ does not have a proper subset.

(v) $A = \{a, b, a, b, a, b,... \}$ is an infinite set. FALSE – The set is $A = \{ a, b \}$ which is a finite set.

(vi) $A = \{a, b, c\}$ and $B = \{1,2,3\}$ are equivalent sets: TRUE$n(A) = 3$ and $n(B) = 3$. Hence $A$ and $B$ are equivalent.

(vii) A set call have infinitely many subsets: FALSE – Not always necessary. For example, $A = \{a, b, c\}$ will have finite subset.

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Question 2: State whether the following statement are true or false:

(i) $1 \in \{1, 2, 3\}$          (ii) $a \subset \{b, c, a\}$          (iii) $\{a\} \in \{a, b, c\}$

(vi) $\{a, b\} = \{a, a, b, b, a\}$          (v) The set $\{x: x +8 =8\}$ is the null set.

(i) $1 \in \{1, 2, 3\}$: TRUE$1$ is an element of  $\{1, 2, 3\}$

(ii) $a \subset \{b, c, a\}$: FALSE$a$ is an element of  $\{b, c, a\}$ and not a subset. $\{ a \}$ is a subset of $\{b, c, a\}$

(iii) $\{a\} \in \{a, b, c\}$: FALSE$\{a \}$ is a subset of  $\{a, b, c\}$ . $a \in \{a, b, c\}$

(vi) $\{a, b\} = \{a, a, b, b, a\}$: TRUE – Both sets are $\{a, b \}$

(v) The set $\{x: x +8 =8\}$ is the null set : FALSE – The set is $\{ 0 \}$. This is not a null set. It has one element.

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Question 3: Decide among the following sets, which are subsets of which:

$A = \{x: x$ satisfies $x^2-8x +12=0\}$ , $B = \{2, 4, 6\}, C = \{2, 4, 6, 8, \cdots \}, D = \{6\}$

$A = \{x: x$ satisfies $x^2-8x +12=0\} \Rightarrow A = \{ 2, 6 \}$

$B = \{2, 4, 6\}, C = \{2, 4, 6, 8, \cdots \}, D = \{6\}$

Therefore $D \subset A \subset B \subset C$

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Question 4: Write which of the following statements are true? Justify your answer

(i) The set of all integers is contained in the set of all rational numbers.

(ii) The set of all crows is contained in the set of all birds.

(iii) The set of all rectangles is contained in the set of all squares.

(vi) The set of all real numbers is contained in the set of all complex numbers.

(v) The sets $P = \{a\}$ and $B = \{\{a\}$ are equal.

(vi) The sets $A = \{x: x$ is a letter of the word $"LITTLE"\}$ and, $B = \{x: x$ is a letter of the word $"TITLE" \}$ are equal

(i) The set of all integers is contained in the set of all rational numbers: TRUE –  rational number can be written as $\frac{m}{n}$. So, if we were to have $n = 1$, then all rational numbers will be integers.

(ii) The set of all crows is contained in the set of all birds: TRUE – All crows are birds. Hence set of all birds will contain set of all crows.

(iii) The set of all rectangles is contained in the set of all squares: FALSE – Every square can be a rectangle but rectangle cannot be a square.

(iv) The set of all real numbers is contained in the set of all complex numbers: TRUE –  Every real number can be written in the form $(a + ib)$. Therefore we can say that the set of all real numbers is contained in the set of all complex numbers.

(v) The sets $P = \{a\}$ and $B = \{\{a\} \}$ are equal: FALSE$B = \{\{a\} \} = \{ P \}$ Therefore $P \neq \{ P \}$

(vi) The sets $A = \{x: x$ is a letter of the word $"LITTLE"\}$ and, $B = \{ x : x$ is a letter of the word $"TITLE" \}$ are equal: TRUE$A = \{ L, I, T, E \}$ and $B = \{ T, I, L, E \}$. They are equal as they have the same elements.

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Question 5: Which of the following statements are correct? Write a correct form of each of the incorrect statements.

(i) $a \subset \{a, b, c\}$     (ii) $a \in \{a, b, c\}$     (iii) $a \in \{\{a\}, b, c\}$     (vi) $\{a\} \subset \{\{a\}, b\}$     (v) $\{b, c\} \subset \{a, \{b, c\}\}$     (vi) $\{ a, b \} \subset \{ a, \{b, c \} \}$    (vii) $\phi \in \{a, b\}$     (viii) $\phi \subset \{a, b, c\}$     (ix) $\{x: x+ 3 = 3 \} = \phi$

(i) $a \subset \{a, b, c\}$: FALSE$a \in \{a, b, c\}$

(ii) $\{ a \} \in \{a, b, c\}$: FALSE – $\{ a \} \subset \{a, b, c\}$

(iii) $a \in \{\{a\}, b, c\}$: FALSE$\{ a \} \in \{\{a\}, b, c\}$

(iv) $\{ a \} \subset \{\{a\}, b\}$: FALSE$\{ a \} \in \{\{a\}, b\}$ or $\{ \{ a \} \} \subset \{\{a\}, b\}$

(v) $\{b, c\} \subset \{a, \{b, c\}\}$: FALSE$\{b, c\} \in \{a, \{b, c\} \}$

(vi) $\{ a, b \} \subset \{ a, \{b, c \} \}$: FALSE$\{ a, b \} \not\subset \{ a, \{b, c \} \}$

(vii) $\phi \in \{a, b\}$; FALSE$\phi \subset \{a, b\}$

(viii) $\phi \subset \{a, b, c\}$: TRUE$\phi$ is a subset of all sets.

(ix) $\{x: x+ 3 = 3 \} = \phi$: FALSE –  $\{x: x+ 3 = 3 \} = \{ 0 \}$

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Question 6: Let $A =\{a,b, \{c,d\}, e\}$. Which of the following statements are false and why?

(i) $\{c, d\} \subset A$     (ii) $\{c, d\} \in A$     (iii) $\{\{c, d\}\} \subset A$     (vi) $a \in A$     (v) $a \subset A$     (vi) $\{a, b, e\} \subset A$     (vii) $\{a, b, e\} \in A$     (viii) $\{a, b, c\} \subset A$     (ix) $\phi \in A$          (x) $\{\phi\} \subset A$

(i) $\{c, d\} \subset A$: FALSEThe correct statement should be $\{c, d\} \in A$

(ii) $\{c, d\} \in A$: TRUE

(iii) $\{\{c, d\}\} \subset A$: TRUE

(vi) $a \in A$: TRUE

(v) $a \subset A$: FALSEThe correct statement should be $a \in A$ or $\{ a \} \subset A$

(vi) $\{a, b, e \} \subset A$: TRUE

(vii) $\{a, b, e \} \in A$: FALSEThe correct statement should be $\{a, b, e \} \subset A$ or $a, b, e \in A$

(viii) $\{a, b, c\} \subset A$: FALSEThe correct statement should be $\{a, b, c\} \not\subset A$

(ix) $\phi \in A$: FALSEThe correct statement should be $\phi \subset A$

(x) $\{\phi\} \subset A$: FALSE – The correct statement should be $\phi \subset A$

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Question 7: Let $A = \{\{1,2,3\},\{4,5\},\{6,7,8\}\}$. Determine which of the following is true or false

(i) $1 \in A$     (ii) $\{1, 2, 3\} \subset A$     (iii) $\{6, 7, 8\} \in A$     (iv) $\{\{4, 5\}\} \subset A$  (v) $\phi \in A$     (vi) $\phi \subset A$

(i) $1 \in A$: FALSEThe correct statement should be $1 \notin A$:

(ii) $\{1, 2, 3\} \subset A$: FALSEThe correct statement should be  $\{1, 2, 3\} \in A$ or $\{ \{1, 2, 3\} \} \subset A$

(iii) $\{6, 7, 8\} \in A$: TRUE

(vi) $\{\{4, 5\}\} \subset A$: TRUE

(v) $\phi \in A$: FALSEThe correct statement should be $\phi \subset A$

(vi) $\phi \subset A$: TRUE

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Question 8: Let $A = \{ \phi , \{\phi\}, 1, \{1, \phi\}, 2\}$. Which of the following is true?

(i) $\phi \in A$     (ii) $\{\phi\} \in A$     (iii) $\{1\} \in A$     (iv) $\{2, \phi\} \subset A$     (v) $2 \subset A$                 (vi) $\{2, \{1\}\} \not\subset A$     (vii) $\{\{2\}, \{1\}\} \not\subset A$     (viii) $\{ \phi , \{\phi\}, 1, \{1, \phi\}\} \subset A$     (xi) $\{\{ \phi \}\} \subset A$

(i) $\phi \in A$: TRUE

(ii) $\{\phi\} \in A$: TRUE

(iii) $\{1\} \in A$: FALSE – The correct statement should be$\{1\} \subset A$

(vi) $\{2, \phi\} \subset A$: TRUE

(v) $2 \subset A$: FALSE – The correct statement should be $2 \in A$

(vi) $\{2, \{1\}\} \not\subset A$: TRUE

(vii) $\{\{2\}, \{1\}\} \not\subset A$: TRUE

(viii) $\{ \phi , \{\phi\}, 1, \{1, \phi\}\} \subset A$: TRUE

(xi) $\{\{ \phi \}\} \subset A$: TRUE

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Question 9: Write down all possible subsets of each of the following sets:

(i) $\{a\}$   (ii) $\{0, 1\}$   (iii) $\{a, b, c\}$   (vi) $\{1, \{1\}\}$   (v) $\{ \phi\}$

(i)  Subsets are $\phi, \{a\}$

(ii) Subsets are $\phi , \{ 0 \}, \{ 1\}, \{0, 1 \}$

(iii)  Subsets are $\phi , \{a\}, \{ b\}, \{ c\}, \{a, b\}, \{b, c\}, \{a, c\}, \{a, b, c\}$

(vi)  Subsets are $\phi , \{ 1 \}, \{ \{1\} \}, \{1, \{1\}\}$

(v)  Subsets are $\phi , \{ \phi\}$

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Question 10: Write down all possible proper subsets each of the following sets:

(i) $\{1,2\}$      (ii) $\{1, 2, 3\}$      (iii) $\{1\}$

(i) Proper Subsets are $\phi , \{1 \}, \{ 2\}$

(ii) Proper Subsets are $\phi ,\{1 \}, \{ 2 \}, \{ 3\}, \{1, 2 \}, \{ 2, 3\}, \{1, 3\}$

(iii) Proper Subsets are $\phi$

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Question 11: What is the total number of proper subsets of a set consisting of $n$ elements?

The total number of subsets of a finite set containing $n$ elements is $2^n$.

Therefore the total number of proper subsets of a finite set containing $n$ elements is $2^n - 1$. Basically we remove the set itself from the list of possible sets.

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Question 12: If $A$ is any set, prove that: $A \subseteq \phi \Leftrightarrow A = \phi$

Since $A$ is a subset of $\phi$ therefore it will contain only the elements of $\phi$. Since $\phi$ possess no elements therefore $A$ also do not possess any element

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Question 13: Prove that $A \subseteq B, B \subseteq C$ and $C \subseteq A \Rightarrow A = C$

Since $A \subseteq B \Rightarrow \forall x \in A \Rightarrow x \in B$

Also since $B \subseteq C \Rightarrow x \in B \Rightarrow x \in C$

$\therefore$ For all $\forall x \in C \Rightarrow x \in A \Rightarrow C \subseteq A$

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Question 14: How many elements has $P (A)$, if $A = \phi$ ?

If $A= \phi$  that means $A$ does not contain any element i.e., $n=0$. Now, number of elements in a power set is $2^n$. $\therefore, n \{ P(A) \} =2^0=1$ Therefore $P(A)$ contains $1$ element.

In other words $\{ A \} = \{ \phi \}$. Which means that $P(A) = 1$.

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Question 15: What universal set (s) would you propose for each of the following: (i) The set of right triangles. (ii) The set of isosceles triangles.

(i) Set of all triangles in a plane.

(ii) Set of all triangles in a plane.

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Question 16: If $X = \{8^n - 7n - 1: n \in N\}$ and $Y = \{49 (n-1): n \in N\}$ , then prove that $X \subseteq Y$.

For $n = 1, x_1 = 8 - 7 - 1 = 0$

For $n \geq 2$

$x_n = 8^n - 7 n + 1$

$= (1+7)^n - 7n - 1$

$= ^nC_0 + ^nC_1 (7)^1 + ^nC_2 (7)^2 + ^nC_3 (7)^3 + \cdots + ^nC_{n-1} 7^{n-1} + ^nC_n (7)^n - 7n - 1$

$= 1 +7n + ^nC_2 (7)^2 + ^nC_3 (7)^3 + \cdots + ^nC_{n-1} 7^{n-1} + ^nC_n (7)^n - 7n - 1$

$= ^nC_2 (7)^2 + ^nC_3 (7)^3 + \cdots + ^nC_{n-1} 7^{n-1} + ^nC_n (7)^n$

$= 7^2 \{ ^nC_2 + ^nC_3 (7)^1 + \cdots + ^nC_{n-1} 7^{n-3} + ^nC_n (7)^{n-2} \}$

$= 49 \{ ^nC_2 + ^nC_3 (7)^1 + \cdots + ^nC_{n-1} 7^{n-3} + ^nC_n (7)^{n-2} \}$

Thus, $x_n$ is some positive integral multiple of 49 for all $n \geq 2$$X$ consists of all positive integral multiple of $49$ that are of the form $= 49 \{ ^nC_2 + ^nC_3 (7)^1 + \cdots + ^nC_{n-1} 7^{n-3} + ^nC_n (7)^{n-2} \}$ along with $0$. $Y = \{49 (n-1): n \in N\}$ implies that it consists of all integral multiples of 49 along with $0$. Therefore \$latex $X \subseteq Y$.

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