Classes
Class 9Class 10First YearSecond Year
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)

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

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$P(B)=\{\Phi\{1\}\{2\}\{3\}\{12\}\{13\}\{23\}\{123\}\}$

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