\displaystyle \textbf{Question 1: }\text{Find the component statements of the following compound statements.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) The sky is blue and the grass is green.}
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ The sky is blue.}
\displaystyle q:\text{ The grass is green.}

\displaystyle \text{(ii) The earth is round or the sun is cold.}
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ The earth is round.}
\displaystyle q:\text{ The sun is cold.}

\displaystyle \text{(iii) All rational numbers are real and all real numbers are complex.}
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ All rational numbers are real.}
\displaystyle q:\text{ All real numbers are complex.}

\displaystyle \text{(iv) }25\text{ is a multiple of }5\text{ and }8.
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ }25\text{ is a multiple of }5.
\displaystyle q:\text{ }25\text{ is a multiple of }8.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{For each of the following statements, determine whether an}
\displaystyle \text{inclusive ``OR'' or exclusive ``OR'' is used. Give reasons for your answer.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) Students can take Hindi or Sanskrit as their third language.}
\displaystyle \text{Exclusive ``OR'' is used because a student can choose either Hindi or Sanskrit,}
\displaystyle \text{but not both as the third language.}

\displaystyle \text{(ii) To enter a country, you need a passport or a voter registration card.}
\displaystyle \text{Inclusive ``OR'' is used because either document is sufficient, and a person}
\displaystyle \text{may also possess both a passport and a voter registration card.}

\displaystyle \text{(iii) A lady gives birth to a baby boy or a baby girl.}
\displaystyle \text{Exclusive ``OR'' is used because the baby is either a boy or a girl,}
\displaystyle \text{but not both.}

\displaystyle \text{(iv) To apply for a driving licence, you should have a ration card or a passport.}
\displaystyle \text{Inclusive ``OR'' is used because either document is acceptable, and a person}
\displaystyle \text{may possess both a ration card and a passport.}
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Write the component statements of the following compound statements}
\displaystyle \text{and check whether the compound statement is true or false.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) To enter a public library, children need an identity card from the school}
\displaystyle \text{or a letter from the school authorities.}
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ To enter a public library, children need an identity card from the school.}
\displaystyle q:\text{ To enter a public library, children need a letter from the school authorities.}
\displaystyle \text{The connective is ``OR''. Assuming this is the library rule, the compound statement is true.}

\displaystyle \text{(ii) All rational numbers are real and all real numbers are not complex.}
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ All rational numbers are real.}
\displaystyle q:\text{ All real numbers are not complex.}
\displaystyle p\text{ is true and }q\text{ is false since every real number is a complex number.}
\displaystyle \text{The connective is ``AND''. Hence the compound statement is false.}

\displaystyle \text{(iii) Square of an integer is positive or negative.}
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ Square of an integer is positive.}
\displaystyle q:\text{ Square of an integer is negative.}
\displaystyle p\text{ is false since }0^2=0,\text{ which is neither positive nor negative.}
\displaystyle q\text{ is also false since the square of an integer can never be negative.}
\displaystyle \text{The connective is ``OR''. Hence the compound statement is false.}

\displaystyle \text{(iv) }x=2\text{ and }x=3\text{ are the roots of }3x^2-x-10=0.
\displaystyle \text{The component statements are:}
\displaystyle p:x=2\text{ is a root of }3x^2-x-10=0.
\displaystyle q:x=3\text{ is a root of }3x^2-x-10=0.
\displaystyle p\text{ is true since }3(2)^2-2-10=0.
\displaystyle q\text{ is false since }3(3)^2-3-10=14\neq0.
\displaystyle \text{The connective is ``AND''. Hence the compound statement is false.}

\displaystyle \text{(v) The sand heats up quickly in the sun and does not cool down fast at night.}
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ The sand heats up quickly in the sun.}
\displaystyle q:\text{ The sand does not cool down fast at night.}
\displaystyle p\text{ is true and }q\text{ is false because sand cools down quickly at night.}
\displaystyle \text{The connective is ``AND''. Hence the compound statement is false.}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Determine whether the following compound statements are true or false.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) Delhi is in India and }2+2=4.
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ Delhi is in India.}
\displaystyle q:2+2=4.
\displaystyle p\text{ and }q\text{ are both true. Since the connective is ``AND'', the compound statement is true.}

\displaystyle \text{(ii) Delhi is in England and }2+2=4.
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ Delhi is in England.}
\displaystyle q:2+2=4.
\displaystyle p\text{ is false and }q\text{ is true. Since the connective is ``AND'', the compound statement is false.}

\displaystyle \text{(iii) Delhi is in India and }2+2=5.
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ Delhi is in India.}
\displaystyle q:2+2=5.
\displaystyle p\text{ is true and }q\text{ is false. Since the connective is ``AND'', the compound statement is false.}

\displaystyle \text{(iv) Delhi is in England and }2+2=5.
\displaystyle \text{The component statements are:}
\displaystyle p:\text{ Delhi is in England.}
\displaystyle q:2+2=5.
\displaystyle p\text{ and }q\text{ are both false. Since the connective is ``AND'', the compound statement is false.}
\displaystyle \\

 

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