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First Year Math Sets, Functions & Groups


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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.
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.Showthatthesetconsistingofelementsoftheforma+3b(abbeingrational)isanabeliangroupw.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.Writetwopropersubsetsofeachofthefollowingsets:vii)W4. 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.Determinewhethereachofthefollowingisatautologyacontingencyoranabsurdity:ii)p(qp)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.Writetwopropersubsetsofeachofthefollowingsets:ii){01}4. Write two proper subsets of each of the following sets: -ii) \{01\}

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