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First Year Math Sets, Functions & Groups 1. Write the converse inverse and contrapositive of the following conditionals: -iii) -p \rightarrow \sim q


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1. Write the converse inverse and contrapositive of the following conditionals: -iii) -p \rightarrow \sim q

7. Show that the set consisting of elements of the form a+\sqrt{3} b ( a b being rational) is an abelian group w.r.t. addition.
7. Show that the set consisting of elements of the form  a+\sqrt{3} b  (  a b  being rational) is an abelian group w.r.t. addition.

7. Show that the set consisting of elements of the form a+\sqrt{3} b ( a b being rational) is an abelian group w.r.t. addition.

Example 1: Ordinary addition multiplication are operations on N . i.e. N is closed with respect to ordinary addition and multiplication because\[\forall a b \in N a+b \in N \wedge a . b \in N\]( \forall stands for" for all" and \wedge stands for" and")
Example 1:    Ordinary addition multiplication are operations on  N .  i.e.  N  is closed with respect to ordinary addition and multiplication because\[\forall a b \in N a+b \in N \wedge a . b \in N\](  \forall  stands for" for all" and  \wedge  stands for" and")

Example 1: Ordinary addition multiplication are operations on N . i.e. N is closed with respect to ordinary addition and multiplication because\[\forall a b \in N a+b \in N \wedge a . b \in N\]( \forall stands for" for all" and \wedge stands for" and")

4. Write two proper subsets of each of the following sets: -vii) W
4. Write two proper subsets of each of the following sets: -vii)  W

4. Write two proper subsets of each of the following sets: -vii) W

4. Determine whether each of the following is a tautology a contingency or an absurdity: -ii) p \rightarrow(q \rightarrow p)
4. Determine whether each of the following is a tautology a contingency or an absurdity: -ii)  p \rightarrow(q \rightarrow p)

4. Determine whether each of the following is a tautology a contingency or an absurdity: -ii) p \rightarrow(q \rightarrow p)

Example 4: If B=\{123\} then\[P(B)=\{\Phi\{1\}\{2\}\{3\}\{12\}\{13\}\{23\}\{123\}\}\]
Example 4: If  B=\{123\}  then\[P(B)=\{\Phi\{1\}\{2\}\{3\}\{12\}\{13\}\{23\}\{123\}\}\]

Example 4: If B=\{123\} then\[P(B)=\{\Phi\{1\}\{2\}\{3\}\{12\}\{13\}\{23\}\{123\}\}\]

4. Write two proper subsets of each of the following sets: -ii) \{01\}
4. Write two proper subsets of each of the following sets: -ii)  \{01\}

4. Write two proper subsets of each of the following sets: -ii) \{01\}

4. Write two proper subsets of each of the following sets: -viii) \{x \mid x \in Q \wedge 0<x \leq 2\}
4. Write two proper subsets of each of the following sets: -viii)  \{x \mid x \in Q \wedge 0<x \leq 2\}

4. Write two proper subsets of each of the following sets: -viii) \{x \mid x \in Q \wedge 0<x \leq 2\}

Example 5:i) \forall x \in A p(x) is true.(To be read: For all x belonging to A the statement p(x) is true).
Example 5:i)  \forall x \in A p(x)  is true.(To be read: For all  x  belonging to  A  the statement  p(x)  is true).

Example 5:i) \forall x \in A p(x) is true.(To be read: For all x belonging to A the statement p(x) is true).

5. Using the Venn diagrams if necessary find the single sets equal to the following: -iv) A \cup \Phi
5. Using the Venn diagrams if necessary find the single sets equal to the following: -iv)  A \cup \Phi

5. Using the Venn diagrams if necessary find the single sets equal to the following: -iv) A \cup \Phi

Example 5: It can be easily verified that ordinary multiplication (but not addition) is an operation on the set \left\{1 \omega \omega^{2}\right\} where \omega^{3}=1 . The adjoining table may be used for the verification of this fact.\begin{tabular}{|c|c|c|c|}\hline \otimes & 1 & \omega & \omega^{2} \\\hline 1 & 1 & \omega & \omega^{2} \\\hline \omega & \omega & \omega^{2} & 1 \\\hline \omega^{2} & \omega^{2} & 1 & \omega \\\hline\end{tabular} ( \omega is pronounced omega)
Example 5: It can be easily verified that ordinary multiplication (but not addition) is an operation on the set  \left\{1 \omega \omega^{2}\right\}  where  \omega^{3}=1 . The adjoining table may be used for the verification of this fact.\begin{tabular}{|c|c|c|c|}\hline \otimes  & 1 &  \omega  &  \omega^{2}  \\\hline 1 & 1 &  \omega  &  \omega^{2}  \\\hline \omega  &  \omega  &  \omega^{2}  & 1 \\\hline \omega^{2}  &  \omega^{2}  & 1 &  \omega  \\\hline\end{tabular} (  \omega  is pronounced omega)

Example 5: It can be easily verified that ordinary multiplication (but not addition) is an operation on the set \left\{1 \omega \omega^{2}\right\} where \omega^{3}=1 . The adjoining table may be used for the verification of this fact.\begin{tabular}{|c|c|c|c|}\hline \otimes & 1 & \omega & \omega^{2} \\\hline 1 & 1 & \omega & \omega^{2} \\\hline \omega & \omega & \omega^{2} & 1 \\\hline \omega^{2} & \omega^{2} & 1 & \omega \\\hline\end{tabular} ( \omega is pronounced omega)

2. Write each of the following sets in the descriptive and tabular forms: -v) \{x \mid x \in P \wedge x<12\}
2. Write each of the following sets in the descriptive and tabular forms: -v)  \{x \mid x \in P \wedge x<12\}

2. Write each of the following sets in the descriptive and tabular forms: -v) \{x \mid x \in P \wedge x<12\}

1. Exhibit A \cup B and A \cap B by Venn diagrams in the following cases: -iii) A \cup A^{\prime}
1. Exhibit  A \cup B  and  A \cap B  by Venn diagrams in the following cases: -iii)   A \cup A^{\prime}

1. Exhibit A \cup B and A \cap B by Venn diagrams in the following cases: -iii) A \cup A^{\prime}

1. Operation \oplus performed on the two-member set G=\{01\} is shown in the adjoining table. Answer the questions: -ii) What is the inverse of 1 ?\begin{tabular}{|c|c|c|}\hline \oplus & 0 & 1 \\\hline 0 & 0 & 1 \\\hline 1 & 1 & 0 \\\hline\end{tabular}
1. Operation  \oplus  performed on the two-member set  G=\{01\}  is shown in the adjoining table. Answer the questions: -ii) What is the inverse of 1 ?\begin{tabular}{|c|c|c|}\hline \oplus  & 0 & 1 \\\hline 0 & 0 & 1 \\\hline 1 & 1 & 0 \\\hline\end{tabular}

1. Operation \oplus performed on the two-member set G=\{01\} is shown in the adjoining table. Answer the questions: -ii) What is the inverse of 1 ?\begin{tabular}{|c|c|c|}\hline \oplus & 0 & 1 \\\hline 0 & 0 & 1 \\\hline 1 & 1 & 0 \\\hline\end{tabular}

2. Repeat Q-1 when A=\mathbb{R}_{1} the set of real numbers. Which of the real lines are functions.
2. Repeat  Q-1  when  A=\mathbb{R}_{1}  the set of real numbers. Which of the real lines are functions.

2. Repeat Q-1 when A=\mathbb{R}_{1} the set of real numbers. Which of the real lines are functions.

6. Supply the missing elements of the third row of the given table so that the operation ※ may be associative.\begin{tabular}{|c|c|c|c|c|}\hline ж & a & b & c & d \\\hline a & a & b & c & d \\\hline b & b & a & c & d \\\hline c & - & - & - & - \\\hline d & d & c & c & d \\\hline\end{tabular}
6. Supply the missing elements of the third row of the given table so that the operation  ※  may be associative.\begin{tabular}{|c|c|c|c|c|}\hline ж &  a  &  b  &  c  &  d  \\\hline a  &  a  &  b  &  c  &  d  \\\hline b  &  b  &  a  &  c  &  d  \\\hline c  &  -  &  -  &  -  &  -  \\\hline d  &  d  &  c  &  c  &  d  \\\hline\end{tabular}

6. Supply the missing elements of the third row of the given table so that the operation ※ may be associative.\begin{tabular}{|c|c|c|c|c|}\hline ж & a & b & c & d \\\hline a & a & b & c & d \\\hline b & b & a & c & d \\\hline c & - & - & - & - \\\hline d & d & c & c & d \\\hline\end{tabular}

3. Under what conditions on A and B are the following statements true? viii) n(A \cap B)=0
3. Under what conditions on  A  and  B  are the following statements true? viii)  n(A \cap B)=0

3. Under what conditions on A and B are the following statements true? viii) n(A \cap B)=0

4. Let U=\{12345678910\} A=\{246810\} B=\{12345\} and C=\{13579\} List the members of each of the following sets: -iii) A \cup B
4. Let  U=\{12345678910\}  A=\{246810\}  B=\{12345\}  and  C=\{13579\}  List the members of each of the following sets: -iii)  A \cup B

4. Let U=\{12345678910\} A=\{246810\} B=\{12345\} and C=\{13579\} List the members of each of the following sets: -iii) A \cup B

7. If U=\{12345 \ldots 20\} and A=\{135 \ldots . 19) verify the following:-ii) A \cap U=A
7. If  U=\{12345 \ldots 20\}  and  A=\{135 \ldots . 19)  verify the following:-ii)  A \cap U=A

7. If U=\{12345 \ldots 20\} and A=\{135 \ldots . 19) verify the following:-ii) A \cap U=A

1. Write the following sets in set builder notation:vii) \{Peshawar Lahore Karachi Quetta\}
1. Write the following sets in set builder notation:vii) \{Peshawar Lahore Karachi Quetta\}
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1. Write the following sets in set builder notation:vii) \{Peshawar Lahore Karachi Quetta\}

3. Show that each of the following statements is a tautology: -ii) p \rightarrow(p \vee q)
3. Show that each of the following statements is a tautology: -ii)   p \rightarrow(p \vee q)
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3. Show that each of the following statements is a tautology: -ii) p \rightarrow(p \vee q)

3. Which of the following sets are finite and which of these are infinite?xi) \{1234 \ldots\}
3. Which of the following sets are finite and which of these are infinite?xi)  \{1234 \ldots\}
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3. Which of the following sets are finite and which of these are infinite?xi) \{1234 \ldots\}

5. Which of the following binary operations shown in tables (a) and (b) is commutative?\begin{tabular}{|c|c|c|c|c|}\hline 萬 & a & b & c & d \\\hline a & a & c & b & d \\\hline b & b & c & b & a \\\hline c & c & d & b & c \\\hline d & a & a & b & b \\\hline\end{tabular}\begin{tabular}{|c|c|c|c|c|}\hline 䒝 & a & b & c & d \\\hline a & a & c & b & d \\\hline b & c & d & b & a \\\hline c & b & b & a & c \\\hline d & d & a & c & d \\\hline\end{tabular}(a)(b)
5. Which of the following binary operations shown in tables (a) and (b) is commutative?\begin{tabular}{|c|c|c|c|c|}\hline 萬 &  a  &  b  &  c  &  d  \\\hline a  &  a  &  c  &  b  &  d  \\\hline b  &  b  &  c  &  b  &  a  \\\hline c  &  c  &  d  &  b  &  c  \\\hline d  &  a  &  a  &  b  &  b  \\\hline\end{tabular}\begin{tabular}{|c|c|c|c|c|}\hline 䒝 &  a  &  b  &  c  &  d  \\\hline a  &  a  &  c  &  b  &  d  \\\hline b  &  c  &  d  &  b  &  a  \\\hline c  &  b  &  b  &  a  &  c  \\\hline d  &  d  &  a  &  c  &  d  \\\hline\end{tabular}(a)(b)
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5. Which of the following binary operations shown in tables (a) and (b) is commutative?\begin{tabular}{|c|c|c|c|c|}\hline 萬 & a & b & c & d \\\hline a & a & c & b & d \\\hline b & b & c & b & a \\\hline c & c & d & b & c \\\hline d & a & a & b & b \\\hline\end{tabular}\begin{tabular}{|c|c|c|c|c|}\hline 䒝 & a & b & c & d \\\hline a & a & c & b & d \\\hline b & c & d & b & a \\\hline c & b & b & a & c \\\hline d & d & a & c & d \\\hline\end{tabular}(a)(b)

4. Let U=\{12345678910\} A=\{246810\} B=\{12345\} and C=\{13579\} List the members of each of the following sets: -vi) A^{c} \cup C^{c}
4. Let  U=\{12345678910\}  A=\{246810\}  B=\{12345\}  and  C=\{13579\}  List the members of each of the following sets: -vi)  A^{c} \cup C^{c}
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4. Let U=\{12345678910\} A=\{246810\} B=\{12345\} and C=\{13579\} List the members of each of the following sets: -vi) A^{c} \cup C^{c}

6. Taking any set say A=\{12345\} verify the following: -i) A \cup \Phi=A
6. Taking any set say  A=\{12345\}  verify the following: -i)  A \cup \Phi=A
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6. Taking any set say A=\{12345\} verify the following: -i) A \cup \Phi=A

6. Taking any set say A=\{12345\} verify the following: -ii) A \cup A=A
6. Taking any set say  A=\{12345\}  verify the following: -ii)  A \cup A=A
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6. Taking any set say A=\{12345\} verify the following: -ii) A \cup A=A

Example 8: For a set A of distinct elements the binary operation * on A defined by\[x \text { 䒝 } y=x \forall x y \in A\]is non commutative and assocaitve.
Example  8:  For a set  A  of distinct elements the binary operation  *  on  A  defined by\[x \text { 䒝 } y=x \forall x y \in A\]is non commutative and assocaitve.
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Example 8: For a set A of distinct elements the binary operation * on A defined by\[x \text { 䒝 } y=x \forall x y \in A\]is non commutative and assocaitve.

7. If U=\{12345 \ldots . 20\} and A=\{135 \ldots . . 19) verify the following:-iii) A \cap A^{\prime}=\Phi
7. If  U=\{12345 \ldots . 20\}  and  A=\{135 \ldots . . 19)  verify the following:-iii)  A \cap A^{\prime}=\Phi
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7. If U=\{12345 \ldots . 20\} and A=\{135 \ldots . . 19) verify the following:-iii) A \cap A^{\prime}=\Phi

2. Construct truth tables for the following statements: -iii) \sim(p \rightarrow q) \leftrightarrow(p \wedge \sim q)
2. Construct truth tables for the following statements: -iii)   \sim(p \rightarrow q) \leftrightarrow(p \wedge \sim q)
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2. Construct truth tables for the following statements: -iii) \sim(p \rightarrow q) \leftrightarrow(p \wedge \sim q)

1. Write the converse inverse and contrapositive of the following conditionals: -iii) -p \rightarrow \sim q
1. Write the converse inverse and contrapositive of the following conditionals: -iii)  -p \rightarrow \sim q
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1. Write the converse inverse and contrapositive of the following conditionals: -iii) -p \rightarrow \sim q

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