\displaystyle \textbf{1.}\ \text{A binary operation on a set }S\text{ is a function from }S\times S\text{ to }S.
\displaystyle \text{A binary operation }*\text{ on a set }S\text{ associates any two elements }a,b\in S\text{ to a unique element}
\displaystyle a*b\in S.

\displaystyle \textbf{2.}\ \text{A binary operation }*\text{ on a set }S\text{ is said to be}
\displaystyle \text{(i) commutative, if }a*b=b*a\text{ for all }a,b\in S.
\displaystyle \text{(ii) associative, if }(a*b)*c=a*(b*c)\text{ for all }a,b,c\in S.
\displaystyle \text{(iii) distributive over a binary operation on }S,\text{ if }a*(b\circ c)=(a*b)\circ (a*c)
\displaystyle \text{and, }(b\circ c)*a=(b*a)\circ (c*a)\text{ for all }a,b,c\in S.

\displaystyle \textbf{3.}\ \text{Let }*\text{ be a binary operation on a set }S.\text{ An element }e\in S\text{ is said to be identity element for} \\ \text{the binary operation }*,\text{ if }a*e=a=e*a\text{ for all }a\in S.

\displaystyle \textbf{4.}\ \text{Let }*\text{ be a binary operation on a set }S\text{ and }e\in S\text{ be the identity element. An element }a\in S\text{ is } \text{said to be invertible, if there exists an element }b\in S\text{ such that }a*b=e=b*a.

\displaystyle \textbf{5.}\ \text{A binary operation on a finite set can be completely described by means of} \\ \text{composition table.}
\displaystyle \text{From the composition table, we can infer the following properties of the binary operation:}
\displaystyle \text{(i) The binary operation is commutative if the composition table is symmetric about the} \\ \text{leading diagonal.}
\displaystyle \text{(ii) If the row headed by an element say }e\text{ coincides with row at the top and the column}
\displaystyle \text{headed by }e\text{ coincides with the column on the extreme left, then }e\text{ is the identity} \\ \text{element.}
\displaystyle \text{(iii) If each row, except the top-most row, or each column, except the left-most column,}
\displaystyle \text{contains the identity element. Then, every element of the set is invertible with respect}
\displaystyle \text{to the given binary operation.}

\displaystyle \textbf{6.}\ \text{Total number of binary operations on a set consisting of }n\text{ elements is }n^{\,n^{2}}.
\displaystyle \text{Total number of commutative binary operations on a set consisting of }n\text{ elements is }n^{\frac{n(n+1)}{2}}.

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.