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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)

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")

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)

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\}