\displaystyle \textbf{Question 1: }\text{Write down the following statements, using set notation:}

\displaystyle \text{(i) Set }A\text{ is a proper subset of set }B
\displaystyle \text{Answer:}
\displaystyle A\subset B
\displaystyle \text{Note: Let }A\text{ and }B\text{ be any two sets. Then }A\text{ is called a proper subset of }B
\displaystyle \text{if every element of }A\text{ belongs to }B,\text{ and }B\text{ has at least one element that}
\displaystyle \text{does not belong to }A.

\displaystyle \text{(ii) Set }C\text{ is a superset of set }D
\displaystyle \text{Answer:}
\displaystyle C\supseteq D
\displaystyle \text{Note: If }D\text{ is a subset of }C,\text{ then }C\text{ is called a superset of }D.

\displaystyle \text{(iii) Set }B\text{ contains set }A
\displaystyle \text{Answer:}
\displaystyle B\supseteq A
\displaystyle \text{Note: If }B\text{ contains every element of }A,\text{ then }B\text{ is a superset of }A.

\displaystyle \text{(iv) Neither }A\text{ is a subset of }B,\text{ nor }B\text{ is a subset of }A
\displaystyle \text{Answer:}
\displaystyle A\not\subseteq B\text{ and }B\not\subseteq A
\displaystyle \text{Note: If there exists even a single element in }A\text{ that does not belong to }B,\text{ then}
\displaystyle A\text{ is not a subset of }B.\text{ Similarly, if there is any element in }B\text{ that does not}
\displaystyle \text{belong to }A,\text{ then }B\text{ is not a subset of }A.

\displaystyle \\

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

\displaystyle \text{(i) }B\subset C\subset A
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: Not all rectangles are squares and hence }B\text{ is not a proper subset of }C.
\displaystyle \text{All squares are quadrilaterals and hence }C\text{ is a proper subset of }A.

\displaystyle \text{(ii) }C\subset B\subset A
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: All squares are rectangles and hence }C\text{ is a proper subset of }B.
\displaystyle \text{All rectangles are quadrilaterals and hence }B\text{ is a proper subset of }A.

\displaystyle \text{(iii) }C\subset D\subset A
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: All squares are rhombuses and hence }C\text{ is a proper subset of }D.
\displaystyle \text{All rhombuses are quadrilaterals and hence }D\text{ is a proper subset of }A.

\displaystyle \text{(iv) }D\subset C\subset A
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: Not all rhombuses are squares and hence }D\text{ is not a proper subset of }C.
\displaystyle \text{However, all squares are quadrilaterals and hence }C\text{ is a proper subset of }A.

\displaystyle \text{(v) }A\supseteq B\supseteq C
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: }A\text{ is a superset of }B\text{ and }B\text{ is a superset of }C.
\displaystyle \text{This is because every rectangle is a quadrilateral and every square is a rectangle.}

\displaystyle \text{(vi) }A\subseteq B\subseteq C
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: Not every quadrilateral is a rectangle, so }A\not\subseteq B.
\displaystyle \text{Also, not every rectangle is a square, so }B\not\subseteq C.

\displaystyle \\

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

\displaystyle \text{(i) }B\subset C\subset A
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: Not all isosceles triangles are equilateral triangles. Hence }B\not\subset C.
\displaystyle \text{However, all equilateral triangles are triangles and hence }C\subset A.

\displaystyle \text{(ii) }C\subset B\subset A
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: All equilateral triangles are isosceles triangles and hence }C\subset B.
\displaystyle \text{All isosceles triangles are triangles and hence }B\subset A.

\displaystyle \\

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

\displaystyle \text{(i) }1\subset A
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: }1\text{ is an element of }A,\text{ not a set. Therefore, }1\in A.

\displaystyle \text{(ii) }\{1\}\in A
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: The elements of }A\text{ are }1\text{ and }2,\text{ not the set }\{1\}.
\displaystyle \text{Therefore, }\{1\}\notin A.\text{ However, }\{1\}\subset A.

\displaystyle \text{(iii) }1\in A
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: }1\text{ is an element of set }A.

\displaystyle \text{(iv) }\phi\in A
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: }\phi\text{ is the empty set and is not an element of }A=\{1,2\}.

\displaystyle \text{(v) }\phi\subset A
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: The empty set is a subset of every set. Hence }\phi\subset A.

\displaystyle \text{(vi) }\{1\}\notin A
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: }\{1\}\text{ is not an element of }A.\text{ The element }1\text{ belongs to }A.
\displaystyle \text{However, }\{1\}\text{ is a subset of }A,\text{ so }\{1\}\subset A.

\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Which of the following statements are correct?}

\displaystyle \text{(i) }a\subset\{a,b,c\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: }a\text{ is an element of the set }\{a,b,c\},\text{ not a set. Hence it should be}
\displaystyle a\in\{a,b,c\}.

\displaystyle \text{(ii) }\{a\}\in\{a,b,c\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: }\{a\}\text{ is not an element of }\{a,b,c\}.\text{ However, }\{a\}\text{ is a subset of}
\displaystyle \{a,b,c\}.\text{ Hence }\{a\}\subset\{a,b,c\}.

\displaystyle \text{(iii) }\{a\}\subset\{a,b,c\}
\displaystyle \text{Answer:}
\displaystyle \text{Correct}
\displaystyle \text{Note: }\{a\}\text{ is a proper subset of }\{a,b,c\}\text{ because every element of }\{a\}
\displaystyle \text{belongs to }\{a,b,c\},\text{ and the two sets are not equal.}

\displaystyle \text{(iv) }\phi\in\{a,b,c\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: }\phi\text{ is not an element of }\{a,b,c\}.\text{ However, the empty set is a subset}
\displaystyle \text{of every set. Hence }\phi\subset\{a,b,c\}.

\displaystyle \text{(v) }\phi\subset\{a,b,c\}
\displaystyle \text{Answer:}
\displaystyle \text{Correct}
\displaystyle \text{Note: The empty set is a subset of every set. Hence }\phi\subset\{a,b,c\}.

\displaystyle \text{(vi) }\{\phi\}\subset\{a,b,c\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: The set }\{\phi\}\text{ contains one element, namely }\phi.\text{ Since }\phi\notin\{a,b,c\},
\displaystyle \{\phi\}\text{ is not a subset of }\{a,b,c\}.

\displaystyle \text{(vii) }a\in\{\{a\},b\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: The elements of }\{\{a\},b\}\text{ are }\{a\}\text{ and }b,\text{ not }a.
\displaystyle \text{Therefore, }a\notin\{\{a\},b\},\text{ but }\{a\}\in\{\{a\},b\}.

\displaystyle \text{(viii) }\{a\}\subseteq\{\{a\},b\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: For }\{a\}\text{ to be a subset of }\{\{a\},b\},\text{ its element }a\text{ must belong to}
\displaystyle \{\{a\},b\}.\text{ But }a\notin\{\{a\},b\}.\text{ Hence }\{a\}\not\subseteq\{\{a\},b\}.
\displaystyle \text{However, }\{a\}\in\{\{a\},b\}.

\displaystyle \text{(ix) }\{a,b\}\in\{\{a,b\},c\}
\displaystyle \text{Answer:}
\displaystyle \text{Correct}
\displaystyle \text{Note: The elements of }\{\{a,b\},c\}\text{ are }\{a,b\}\text{ and }c.
\displaystyle \text{Therefore, }\{a,b\}\in\{\{a,b\},c\}.

\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Which of the following statements are true?}

\displaystyle \text{(i) }\phi=\{0\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: }\{0\}\text{ contains one element, namely }0.\text{ Therefore, it is not an empty set.}
\displaystyle \text{Hence }\phi\neq\{0\}.

\displaystyle \text{(ii) }\phi=\{\phi\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: }\phi\text{ has no elements, whereas }\{\phi\}\text{ contains one element, namely }\phi.
\displaystyle \text{Therefore, }\phi\neq\{\phi\}.

\displaystyle \text{(iii) }\phi\in\{0\}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: The only element of }\{0\}\text{ is }0.\text{ Hence }\phi\notin\{0\}.

\displaystyle \text{(iv) }\phi\in\{\phi\}
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: }\{\phi\}\text{ contains exactly one element, namely }\phi.
\displaystyle \text{Hence }\phi\in\{\phi\}.

\displaystyle \text{(v) }\phi\in\{\phi,\{0\}\}
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: The elements of }\{\phi,\{0\}\}\text{ are }\phi\text{ and }\{0\}.
\displaystyle \text{Hence }\phi\in\{\phi,\{0\}\}.

\displaystyle \text{(vi) }\phi\subset\{0\}
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: The empty set is a subset of every set. Since }\phi\neq\{0\},\text{ it is also a proper}
\displaystyle \text{subset of }\{0\}.\text{ Hence }\phi\subset\{0\}.

\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Which of the following statements are true?}

\displaystyle \text{(i) For any two sets }A\text{ and }B,\text{ either }A\subseteq B\text{ or }B\subseteq A
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: For }A\text{ to be a subset of }B,\text{ every element of }A\text{ must belong to }B.
\displaystyle \text{This need not be true, as }A\text{ can have elements that are not in }B,\text{ and }B\text{ can have}
\displaystyle \text{elements that are not in }A.\text{ Hence it is not necessary that either }A\subseteq B\text{ or }B\subseteq A.

\displaystyle \text{(ii) Every subset of a finite set is a finite set}
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: A finite set contains a finite number of elements. Any subset can contain only elements}
\displaystyle \text{from the original set. Therefore, every subset of a finite set must also be finite.}

\displaystyle \text{(iii) Every subset of an infinite set is infinite}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: Let }A=\{1,2,3,4,5,\ldots\}.\text{ Then }\{1\}\subset A.
\displaystyle \text{Although }A\text{ is infinite, its subset }\{1\}\text{ is finite.}

\displaystyle \text{(iv) Every set has a proper subset}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: The empty set has no proper subset because its only subset is the empty set itself.}
\displaystyle \text{Every non-empty set has the empty set as a proper subset.}

\displaystyle \text{(v) If }A\text{ has }n\text{ elements, then }P(A)\text{ has }2^n\text{ elements}
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: The set of all possible subsets of a set }A\text{ is called the power set of }A
\displaystyle \text{and is denoted by }P(A).\text{ If }A\text{ contains }n\text{ elements, then }P(A)\text{ contains }2^n
\displaystyle \text{elements, each of which is a subset of }A.

\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Let }A\text{ be the set of letters in the word 'seed'. Find:}

\displaystyle \text{(i) }A
\displaystyle \text{Answer:}
\displaystyle A=\{s,e,d\}

\displaystyle \text{(ii) }n(A)
\displaystyle \text{Answer:}
\displaystyle n(A)=3

\displaystyle \text{(iii) Number of subsets of }A
\displaystyle \text{Answer:}
\displaystyle 2^3=8
\displaystyle \text{Note: The set of all possible subsets of a set }A\text{ is called the power set of }A
\displaystyle \text{and is denoted by }P(A).\text{ If }A\text{ contains }n\text{ elements, then }P(A)\text{ contains }2^n
\displaystyle \text{elements, each of which is a subset of }A.

\displaystyle \text{(iv) Number of proper subsets of }A
\displaystyle \text{Answer:}
\displaystyle 2^3-1=7
\displaystyle \text{Note: A set containing }n\text{ elements has }2^n-1\text{ proper subsets.}

\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Find all possible subsets of each of the following sets:}

\displaystyle \text{(i) }A=\{4,9\}
\displaystyle \text{Answer:}
\displaystyle \phi,\{4\},\{9\},\{4,9\}
\displaystyle \text{Note: Number of elements }=2.\text{ Total number of subsets }=2^2=4.

\displaystyle \text{(ii) }B=\{2,3,8\}
\displaystyle \text{Answer:}
\displaystyle \phi,\{2\},\{3\},\{8\},\{2,3\},\{2,8\},\{3,8\},\{2,3,8\}
\displaystyle \text{Note: Number of elements }=3.\text{ Total number of subsets }=2^3=8.

\displaystyle \text{(iii) }C=\{0,1,2\}
\displaystyle \text{Answer:}
\displaystyle \phi,\{0\},\{1\},\{2\},\{0,1\},\{0,2\},\{1,2\},\{0,1,2\}
\displaystyle \text{Note: Number of elements }=3.\text{ Total number of subsets }=2^3=8.

\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Find the power set of each of the following:}

\displaystyle \text{(i) }A=\{0,5\}
\displaystyle \text{Answer:}
\displaystyle P(A)=\{\phi,\{0\},\{5\},\{0,5\}\}
\displaystyle \text{Note: }P(A)\text{ is the set of all possible subsets of }A.

\displaystyle \text{(ii) }B=\{7,9\}
\displaystyle \text{Answer:}
\displaystyle P(B)=\{\phi,\{7\},\{9\},\{7,9\}\}

\displaystyle \text{(iii) }C=\{2,4,6\}
\displaystyle \text{Answer:}
\displaystyle P(C)=\{\phi,\{2\},\{4\},\{6\},\{2,4\},\{2,6\},\{4,6\},\{2,4,6\}\}

\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Let }A=\{1,\{2\}\}.\text{ Find the power set of }A.
\displaystyle \text{Answer:}
\displaystyle P(A)=\{\phi,\{1\},\{\{2\}\},\{1,\{2\}\}\}
\displaystyle \text{Note: The set }A\text{ has two elements, namely }1\text{ and }\{2\}.\text{ Here }\{2\}\text{ is treated}
\displaystyle \text{as a single element of }A.\text{ Hence }n(A)=2\text{ and }P(A)\text{ has }2^2=4\text{ elements.}

\displaystyle \\

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

\displaystyle \text{Answer:}
\displaystyle \xi=\{1,2,3,4,5,\ldots,49\}.
\displaystyle \text{Since }x^2\in\xi,\text{ we require }x^2<50.\text{ Therefore,}
\displaystyle A=\{1,2,3,4,5,6,7\}.
\displaystyle B=\{1,4,9,16,25,36,49\}.
\displaystyle C=\{1,2,3,4,6,9,12,18,36\}.

\displaystyle \text{(i) }A\subseteq B
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: For example, }2\in A\text{ but }2\notin B.\text{ Hence }A\not\subseteq B.

\displaystyle \text{(ii) }A=B
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: The sets do not have the same elements. For example, }2\in A\text{ but }2\notin B.
\displaystyle \text{Hence }A\neq B.

\displaystyle \text{(iii) }A\text{ and }B\text{ are equivalent sets}
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: }n(A)=7\text{ and }n(B)=7.\text{ Hence }A\text{ and }B\text{ are equivalent sets.}

\displaystyle \text{(iv) }B\text{ and }C\text{ are equivalent sets}
\displaystyle \text{Answer:}
\displaystyle \text{False}
\displaystyle \text{Note: }n(B)=7\text{ while }n(C)=9.\text{ Hence }B\text{ and }C\text{ are not equivalent sets.}

\displaystyle \text{(v) }n(A)<n(C)
\displaystyle \text{Answer:}
\displaystyle \text{True}
\displaystyle \text{Note: }n(A)=7\text{ while }n(C)=9.\text{ Therefore, }n(A)<n(C).

\displaystyle \\

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