\displaystyle \textbf{Question 1: } \text{Which of the following statements are true? Give reasons to support}
\displaystyle \text{your answers.}
\displaystyle \text{(i) For any two sets }A\text{ and }B,\text{ either }A\subseteq B\text{ or }B\subseteq A.
\displaystyle \text{(ii) Every subset of an infinite set is infinite.}
\displaystyle \text{(iii) Every subset of a finite set is finite.}
\displaystyle \text{(iv) Every set has a proper subset.}
\displaystyle \text{(v) }\{a,b,a,b,a,b,\ldots\}\text{ is an infinite set.}
\displaystyle \text{(vi) }\{a,b,c\}\text{ and }\{1,2,3\}\text{ are equivalent sets.}
\displaystyle \text{(vii) A set can have infinitely many subsets.}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) FALSE. It is not necessary that one of any two sets is a subset of the other.}
\displaystyle \text{For example, let }A=\{x,y,z\}\text{ and }B=\{a,b,c\}.
\displaystyle \text{Then }A\nsubseteq B\text{ and }B\nsubseteq A.
\displaystyle \text{(ii) FALSE. An infinite set can have a finite subset.}
\displaystyle \text{For example, }\{2\}\subseteq N,\text{ but }\{2\}\text{ is finite and }N\text{ is infinite.}
\displaystyle \text{(iii) TRUE. A subset of a finite set cannot contain more elements than the set itself.}
\displaystyle \therefore \text{Every subset of a finite set is finite.}
\displaystyle \text{(iv) FALSE. The empty set }\phi\text{ has no proper subset.}
\displaystyle \text{(v) FALSE. Repeated elements are written only once in a set.}
\displaystyle \{a,b,a,b,a,b,\ldots\}=\{a,b\}.
\displaystyle \therefore \text{It is a finite set.}
\displaystyle \text{(vi) TRUE. Let }A=\{a,b,c\}\text{ and }B=\{1,2,3\}.
\displaystyle n(A)=3\text{ and }n(B)=3.
\displaystyle \therefore A\text{ and }B\text{ are equivalent sets.}
\displaystyle \text{(vii) TRUE. An infinite set can have infinitely many subsets.}
\displaystyle \text{For example, }N\text{ has the subsets }\{1\},\{1,2\},\{1,2,3\},\ldots
\displaystyle \therefore N\text{ has infinitely many subsets.}
\displaystyle \\

\displaystyle \textbf{Question 2: } \text{State whether the following statements are true or false.}
\displaystyle \text{(i) }1\in\{1,2,3\}\qquad\text{(ii) }a\subset\{b,c,a\}\qquad\text{(iii) }\{a\}\in\{a,b,c\}
\displaystyle \text{(iv) }\{a,b\}=\{a,a,b,b,a\}\qquad\text{(v) The set }\{x:x+8=8\}\text{ is the null set.}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) TRUE. }1\in\{1,2,3\}\text{ since }1\text{ is an element of the set.}
\displaystyle \text{(ii) FALSE. }a\text{ is an element, not a subset, of }\{b,c,a\}.
\displaystyle \{a\}\subset\{b,c,a\}.
\displaystyle \text{(iii) FALSE. }\{a\}\text{ is a subset of }\{a,b,c\}\text{ but not an element of it.}
\displaystyle a\in\{a,b,c\}.
\displaystyle \text{(iv) TRUE. Repeated elements are counted only once in a set.}
\displaystyle \therefore \{a,b\}=\{a,a,b,b,a\}=\{a,b\}.
\displaystyle \text{(v) FALSE. }x+8=8\Rightarrow x=0.
\displaystyle \therefore \{x:x+8=8\}=\{0\},\text{ which is not the null set.}
\displaystyle \\

\displaystyle \textbf{Question 3: } \text{Decide among the following sets, which are subsets of which.}
\displaystyle A=\{x:x\text{ satisfies }x^2-8x+12=0\},\;B=\{2,4,6\},
\displaystyle C=\{2,4,6,8,\ldots\},\;D=\{6\}
\displaystyle \textbf{Answer:}
\displaystyle x^2-8x+12=0.
\displaystyle (x-2)(x-6)=0.
\displaystyle \therefore x=2\text{ or }x=6.
\displaystyle \therefore A=\{2,6\}.
\displaystyle B=\{2,4,6\},\quad C=\{2,4,6,8,\ldots\},\quad D=\{6\}.
\displaystyle \therefore D\subset A\subset B\subset C.
\displaystyle \\

\displaystyle \textbf{Question 4: } \text{Which of the following statements are true? Justify your answers.}
\displaystyle \text{(i) The set of all integers is contained in the set of all rational numbers.}
\displaystyle \text{(ii) The set of all crows is contained in the set of all birds.}
\displaystyle \text{(iii) The set of all rectangles is contained in the set of all squares.}
\displaystyle \text{(iv) The set of all real numbers is contained in the set of all complex numbers.}
\displaystyle \text{(v) The sets }P=\{a\}\text{ and }B=\{\{a\}\}\text{ are equal.}
\displaystyle \text{(vi) }A=\{x:x\text{ is a letter of the word "LITTLE"}\}\text{ and}
\displaystyle B=\{x:x\text{ is a letter of the word "TITLE"}\}\text{ are equal.}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) TRUE. Every integer }m\text{ can be written as }\frac{m}{1}.
\displaystyle \text{Therefore, every integer is a rational number.}
\displaystyle \therefore Z\subseteq Q.
\displaystyle \text{(ii) TRUE. Every crow is a bird.}
\displaystyle \therefore \text{The set of all crows is contained in the set of all birds.}
\displaystyle \text{(iii) FALSE. Every square is a rectangle, but every rectangle need not be a square.}
\displaystyle \therefore \text{The set of rectangles is not contained in the set of squares.}
\displaystyle \text{(iv) TRUE. Every real number }a\text{ can be written as }a+0i.
\displaystyle \text{Therefore, every real number is a complex number.}
\displaystyle \therefore R\subseteq C.
\displaystyle \text{(v) FALSE. }P=\{a\}\text{ has the element }a,
\displaystyle \text{whereas }B=\{\{a\}\}\text{ has the set }\{a\}\text{ as its element.}
\displaystyle B=\{P\}\text{ and hence }P\ne B.
\displaystyle \text{(vi) TRUE. Repeated letters are written only once in a set.}
\displaystyle A=\{L,I,T,E\}.
\displaystyle B=\{T,I,L,E\}=\{L,I,T,E\}.
\displaystyle \therefore A=B.
\displaystyle \\

\displaystyle \textbf{Question 5: } \text{Which of the following statements are correct? Write a correct form}
\displaystyle \text{of each of the incorrect statements.}
\displaystyle \text{(i) }a\subset\{a,b,c\}\qquad\text{(ii) }a\in\{a,b,c\}\qquad\text{(iii) }a\in\{\{a\},b,c\}
\displaystyle \text{(iv) }\{a\}\subset\{\{a\},b\}\qquad\text{(v) }\{b,c\}\subset\{a,\{b,c\}\}
\displaystyle \text{(vi) }\{a,b\}\subset\{a,\{b,c\}\}\qquad\text{(vii) }\phi\in\{a,b\}
\displaystyle \text{(viii) }\phi\subset\{a,b,c\}\qquad\text{(ix) }\{x:x+3=3\}=\phi
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) FALSE. }a\text{ is an element, not a set.}
\displaystyle \text{The correct statement is }a\in\{a,b,c\}.
\displaystyle \text{(ii) TRUE. }a\text{ is an element of }\{a,b,c\}.
\displaystyle \text{(iii) FALSE. The elements of }\{\{a\},b,c\}\text{ are }\{a\},b\text{ and }c.
\displaystyle \text{The correct statement is }\{a\}\in\{\{a\},b,c\}.
\displaystyle \text{(iv) FALSE. The element of }\{a\}\text{ is }a,\text{ but }a\notin\{\{a\},b\}.
\displaystyle \text{The correct statement is }\{a\}\in\{\{a\},b\}.
\displaystyle \text{Alternatively, }\{\{a\}\}\subset\{\{a\},b\}.
\displaystyle \text{(v) FALSE. Neither }b\text{ nor }c\text{ is an element of }\{a,\{b,c\}\}.
\displaystyle \text{The correct statement is }\{b,c\}\in\{a,\{b,c\}\}.
\displaystyle \text{(vi) FALSE. Although }a\in\{a,\{b,c\}\},\text{ we have }b\notin\{a,\{b,c\}\}.
\displaystyle \text{The correct statement is }\{a,b\}\nsubseteq\{a,\{b,c\}\}.
\displaystyle \text{(vii) FALSE. The empty set is not an element of }\{a,b\}.
\displaystyle \text{The correct statement is }\phi\subset\{a,b\}.
\displaystyle \text{(viii) TRUE. The empty set is a subset of every set.}
\displaystyle \text{(ix) FALSE. }x+3=3\Rightarrow x=0.
\displaystyle \text{The correct statement is }\{x:x+3=3\}=\{0\}.
\displaystyle \\

\displaystyle \textbf{Question 6: } \text{Let }A=\{a,b,\{c,d\},e\}.\text{ Which of the following statements are false}
\displaystyle \text{and why?}
\displaystyle \text{(i) }\{c,d\}\subset A\qquad\text{(ii) }\{c,d\}\in A\qquad\text{(iii) }\{\{c,d\}\}\subset A
\displaystyle \text{(iv) }a\in A\qquad\text{(v) }a\subset A\qquad\text{(vi) }\{a,b,e\}\subset A
\displaystyle \text{(vii) }\{a,b,e\}\in A\qquad\text{(viii) }\{a,b,c\}\subset A
\displaystyle \text{(ix) }\phi\in A\qquad\text{(x) }\{\phi\}\subset A
\displaystyle \textbf{Answer:}
\displaystyle \text{The elements of }A\text{ are }a,\;b,\;\{c,d\}\text{ and }e.
\displaystyle \text{(i) FALSE. The elements }c\text{ and }d\text{ do not individually belong to }A.
\displaystyle \text{The correct statement is }\{c,d\}\in A.
\displaystyle \text{(ii) TRUE. The set }\{c,d\}\text{ is an element of }A.
\displaystyle \text{(iii) TRUE. The only element of }\{\{c,d\}\}\text{ is }\{c,d\},\text{ which belongs to }A.
\displaystyle \text{(iv) TRUE. }a\text{ is an element of }A.
\displaystyle \text{(v) FALSE. }a\text{ is an element and not a set.}
\displaystyle \text{The correct statement is }a\in A\text{ or }\{a\}\subset A.
\displaystyle \text{(vi) TRUE. Each of }a,\;b\text{ and }e\text{ belongs to }A.
\displaystyle \therefore \{a,b,e\}\subset A.
\displaystyle \text{(vii) FALSE. }\{a,b,e\}\text{ is not an element of }A.
\displaystyle \text{The correct statement is }\{a,b,e\}\subset A.
\displaystyle \text{(viii) FALSE. Although }a,b\in A,\text{ we have }c\notin A.
\displaystyle \text{The correct statement is }\{a,b,c\}\nsubseteq A.
\displaystyle \text{(ix) FALSE. The empty set is not an element of }A.
\displaystyle \text{The correct statement is }\phi\subset A.
\displaystyle \text{(x) FALSE. For }\{\phi\}\text{ to be a subset of }A,\text{ we must have }\phi\in A.
\displaystyle \text{But }\phi\notin A.\text{ The correct statement is }\phi\subset A.
\displaystyle \\

\displaystyle \textbf{Question 7: } \text{Let }A=\{\{1,2,3\},\{4,5\},\{6,7,8\}\}.\text{ Determine which of the}
\displaystyle \text{following statements are true or false.}
\displaystyle \text{(i) }1\in A\qquad\text{(ii) }\{1,2,3\}\subset A\qquad\text{(iii) }\{6,7,8\}\in A
\displaystyle \text{(iv) }\{\{4,5\}\}\subset A\qquad\text{(v) }\phi\in A\qquad\text{(vi) }\phi\subset A
\displaystyle \textbf{Answer:}
\displaystyle \text{The elements of }A\text{ are }\{1,2,3\},\;\{4,5\}\text{ and }\{6,7,8\}.
\displaystyle \text{(i) FALSE. }1\text{ is not directly an element of }A.
\displaystyle \text{The correct statement is }1\notin A.
\displaystyle \text{(ii) FALSE. The numbers }1,\;2\text{ and }3\text{ are not individual elements of }A.
\displaystyle \text{The correct statement is }\{1,2,3\}\in A.
\displaystyle \text{Alternatively, }\{\{1,2,3\}\}\subset A.
\displaystyle \text{(iii) TRUE. The set }\{6,7,8\}\text{ is an element of }A.
\displaystyle \text{(iv) TRUE. The only element of }\{\{4,5\}\}\text{ is }\{4,5\},\text{ which belongs to }A.
\displaystyle \text{(v) FALSE. The empty set is not an element of }A.
\displaystyle \text{The correct statement is }\phi\subset A.
\displaystyle \text{(vi) TRUE. The empty set is a subset of every set.}
\displaystyle \\

\displaystyle \textbf{Question 8: } \text{Let }A=\{\phi,\{\phi\},1,\{1,\phi\},2\}.\text{ Which of the following}
\displaystyle \text{statements are true?}
\displaystyle \text{(i) }\phi\in A\qquad\text{(ii) }\{\phi\}\in A\qquad\text{(iii) }\{1\}\in A
\displaystyle \text{(iv) }\{2,\phi\}\subset A\qquad\text{(v) }2\subset A
\displaystyle \text{(vi) }\{2,\{1\}\}\nsubseteq A\qquad\text{(vii) }\{\{2\},\{1\}\}\nsubseteq A
\displaystyle \text{(viii) }\{\phi,\{\phi\},1,\{1,\phi\}\}\subset A\qquad\text{(ix) }\{\{\phi\}\}\subset A
\displaystyle \textbf{Answer:}
\displaystyle \text{The elements of }A\text{ are }\phi,\;\{\phi\},\;1,\;\{1,\phi\}\text{ and }2.
\displaystyle \text{(i) TRUE. The empty set }\phi\text{ is an element of }A.
\displaystyle \text{(ii) TRUE. The set }\{\phi\}\text{ is an element of }A.
\displaystyle \text{(iii) FALSE. The set }\{1\}\text{ is not an element of }A.
\displaystyle \text{However, }\{1\}\subset A\text{ since }1\in A.
\displaystyle \text{(iv) TRUE. Both }2\text{ and }\phi\text{ are elements of }A.
\displaystyle \therefore \{2,\phi\}\subset A.
\displaystyle \text{(v) FALSE. }2\text{ is an element and not a set.}
\displaystyle \text{The correct statement is }2\in A.
\displaystyle \text{(vi) TRUE. Although }2\in A,\text{ we have }\{1\}\notin A.
\displaystyle \therefore \{2,\{1\}\}\nsubseteq A.
\displaystyle \text{(vii) TRUE. Neither }\{2\}\text{ nor }\{1\}\text{ is an element of }A.
\displaystyle \therefore \{\{2\},\{1\}\}\nsubseteq A.
\displaystyle \text{(viii) TRUE. Each element of }\{\phi,\{\phi\},1,\{1,\phi\}\}\text{ belongs to }A.
\displaystyle \therefore \{\phi,\{\phi\},1,\{1,\phi\}\}\subset A.
\displaystyle \text{(ix) TRUE. The only element of }\{\{\phi\}\}\text{ is }\{\phi\},\text{ which belongs to }A.
\displaystyle \therefore \{\{\phi\}\}\subset A.
\displaystyle \\

\displaystyle \textbf{Question 9: } \text{Write down all possible subsets of each of the following sets.}
\displaystyle \text{(i) }\{a\}\qquad\text{(ii) }\{0,1\}\qquad\text{(iii) }\{a,b,c\}
\displaystyle \text{(iv) }\{1,\{1\}\}\qquad\text{(v) }\{\phi\}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) The subsets of }\{a\}\text{ are }\phi\text{ and }\{a\}.
\displaystyle \therefore P(\{a\})=\{\phi,\{a\}\}.
\displaystyle \text{(ii) The subsets of }\{0,1\}\text{ are }\phi,\{0\},\{1\}\text{ and }\{0,1\}.
\displaystyle \therefore P(\{0,1\})=\{\phi,\{0\},\{1\},\{0,1\}\}.
\displaystyle \text{(iii) The subsets of }\{a,b,c\}\text{ are}
\displaystyle \phi,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\text{ and }\{a,b,c\}.
\displaystyle \therefore P(\{a,b,c\})=\{\phi,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}.
\displaystyle \text{(iv) The elements }1\text{ and }\{1\}\text{ are distinct.}
\displaystyle \text{The subsets of }\{1,\{1\}\}\text{ are }\phi,\{1\},\{\{1\}\}\text{ and }\{1,\{1\}\}.
\displaystyle \therefore P(\{1,\{1\}\})=\{\phi,\{1\},\{\{1\}\},\{1,\{1\}\}\}.
\displaystyle \text{(v) The subsets of }\{\phi\}\text{ are }\phi\text{ and }\{\phi\}.
\displaystyle \therefore P(\{\phi\})=\{\phi,\{\phi\}\}.
\displaystyle \\

\displaystyle \textbf{Question 10: } \text{Write down all possible proper subsets of each of the following sets.}
\displaystyle \text{(i) }\{1,2\}\qquad\text{(ii) }\{1,2,3\}\qquad\text{(iii) }\{1\}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) The proper subsets of }\{1,2\}\text{ are }
\displaystyle \phi,\;\{1\},\;\{2\}.
\displaystyle \text{(ii) The proper subsets of }\{1,2,3\}\text{ are }
\displaystyle \phi,\;\{1\},\;\{2\},\;\{3\},\;\{1,2\},\;\{1,3\},\;\{2,3\}.
\displaystyle \text{(iii) The only proper subset of }\{1\}\text{ is }\phi.
\displaystyle \\

\displaystyle \textbf{Question 11: } \text{What is the total number of proper subsets of a set consisting of}
\displaystyle n\text{ elements?}
\displaystyle \textbf{Answer:}
\displaystyle \text{A set containing }n\text{ elements has }2^n\text{ subsets.}
\displaystyle \text{A proper subset is any subset except the set itself.}
\displaystyle \therefore \text{The total number of proper subsets is }2^n-1.
\displaystyle \\

\displaystyle \textbf{Question 12: } \text{If }A\text{ is any set, prove that }A\subseteq\phi\Leftrightarrow A=\phi.
\displaystyle \textbf{Answer:}
\displaystyle \text{First, suppose }A\subseteq\phi.
\displaystyle \text{Since }\phi\text{ has no elements, }A\text{ cannot have any element.}
\displaystyle \therefore A=\phi.
\displaystyle \text{Conversely, suppose }A=\phi.
\displaystyle \text{Every set is a subset of itself.}
\displaystyle \therefore A=\phi\Rightarrow A\subseteq\phi.
\displaystyle \therefore A\subseteq\phi\Leftrightarrow A=\phi.
\displaystyle \\

\displaystyle \textbf{Question 13: } \text{Prove that }A\subseteq B,\;B\subseteq C\text{ and }C\subseteq A\Rightarrow A=C.
\displaystyle \textbf{Answer:}
\displaystyle \text{Given, }A\subseteq B\text{ and }B\subseteq C.
\displaystyle \text{Let }x\in A.
\displaystyle \text{Since }A\subseteq B,\text{ we have }x\in B.
\displaystyle \text{Since }B\subseteq C,\text{ we have }x\in C.
\displaystyle \therefore A\subseteq C.
\displaystyle \text{Also, it is given that }C\subseteq A.
\displaystyle \text{Since }A\subseteq C\text{ and }C\subseteq A,
\displaystyle \therefore A=C.
\displaystyle \\

\displaystyle \textbf{Question 14: } \text{How many elements has }P(A),\text{ if }A=\phi\textbf{?}
\displaystyle \textbf{Answer:}
\displaystyle \text{Since }A=\phi,\text{ the set }A\text{ contains no element.}
\displaystyle \therefore n(A)=0.
\displaystyle \text{The number of elements in the power set of a set having }n\text{ elements is }2^n.
\displaystyle \therefore n(P(A))=2^0=1.
\displaystyle \text{Also, the only subset of }\phi\text{ is }\phi\text{ itself.}
\displaystyle \therefore P(A)=P(\phi)=\{\phi\}.
\displaystyle \therefore P(A)\text{ has }1\text{ element.}
\displaystyle \\

\displaystyle \textbf{Question 15: } \text{What universal set would you propose for each of the following?}
\displaystyle \text{(i) The set of right triangles.}\qquad\text{(ii) The set of isosceles triangles.}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) A suitable universal set is the set of all triangles in a plane.}
\displaystyle \text{(ii) A suitable universal set is the set of all triangles in a plane.}
\displaystyle \\

\displaystyle \textbf{Question 16: } \text{If }X=\{8^n-7n-1:n\in N\}\text{ and }Y=\{49(n-1):n\in N\},
\displaystyle \text{prove that }X\subseteq Y.
\displaystyle \textbf{Answer:}
\displaystyle \text{Let }x_n=8^n-7n-1,\text{ where }n\in N.
\displaystyle \text{For }n=1,
\displaystyle x_1=8-7-1=0=49(1-1).
\displaystyle \therefore x_1\in Y.
\displaystyle \text{For }n\geq2,
\displaystyle x_n=8^n-7n-1
\displaystyle =(1+7)^n-7n-1.
\displaystyle \text{Using the binomial theorem,}
\displaystyle x_n=\left\{{}^nC_0+{}^nC_1(7)+{}^nC_2(7)^2+\cdots+{}^nC_n(7)^n\right\}-7n-1.
\displaystyle =1+7n+{}^nC_2(7)^2+{}^nC_3(7)^3+\cdots+{}^nC_n(7)^n-7n-1.
\displaystyle ={}^nC_2(7)^2+{}^nC_3(7)^3+\cdots+{}^nC_n(7)^n.
\displaystyle =49\left\{{}^nC_2+{}^nC_3(7)+\cdots+{}^nC_n(7)^{n-2}\right\}.
\displaystyle \text{Let }k={}^nC_2+{}^nC_3(7)+\cdots+{}^nC_n(7)^{n-2}.
\displaystyle \text{Since }n\geq2,\;k\text{ is a positive integer.}
\displaystyle \therefore x_n=49k=49\{(k+1)-1\}.
\displaystyle \text{Since }k+1\in N,\text{ we have }x_n\in Y.
\displaystyle \text{Thus, every element of }X\text{ belongs to }Y.
\displaystyle \therefore X\subseteq Y.
\displaystyle \\

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