# Classes

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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.

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

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