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  {\Large{\bf {Assignment Set}}}\\
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  {Dated: 07/02/2019 \& Submission deadline: 1.30 PM,14/02/2019. \\
 Submit in my office within the deadline. You can slip under my office door. Submission after deadline will lead $50\%$ deduction of marks obtained.}
  
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\begin{enumerate}
\item Prove the validity of the following sequents:
\begin{enumerate}
\item $p \to (q \lor r), \neg q, \neg r \vdash \neg p$
\item $p\lor \neg p \vdash \neg(r \to q) \land (r \to q)$
\end{enumerate}
\item Let us suppose we want to prove the proposition $P$. Let  $P_1$ and $P_2$ are two propositions at least one of which is true. Then establish a logical equivalence with the help of these propositions to outline the method of "proof by cases".  {\bf Hint. use $P_{1(2)} \to P$ is true }

Prove the statement: There must be a prime between $n$ and $n!$, where $n$ is an integer greater than $2$.

\item Prove: Let $x\in \mathbb{Z}$. If $x^2-6x+5$ is even, then $x$ is odd. Clearly state the method used for this proof. 

\item Prove that set of rationals is countable.

\item {\it Product of two even numbers is even}

Prove the above statement by contradiction.

Prove the contrapositive of the above proposition. 

\item {\it Every amount of postage that is at least 12 cents can be made from 4-cent and 5-cent stamps}

Give a proof of the proposition. What kind of method have you used here?

\item Prove every subset of a countable set is countable

\item Let $A=[-1,1]$, and let $f: A\to A$, $g: A\to A$, and $h: A\to A$ be functions defined by\\
(i) $f(x)\sin x$, (ii) $g(x)=\sin \pi x$, and (iii) $h(x)=\sin\frac{\pi}{2}x$. \\

Check whether these functions are one-one, onto or bijective.

Let $B=[0,1]$. Is set B is equivalent to A?  

\end{enumerate}
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