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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\}