Number of binary operations on a set with n elements is ${{n}^{{{n}^{2}}}}$ Here, S = {a, b, c} Number of elements in S = 3 Number of binary operations on a set with 3 elements is...

LCM 1 2 3 4 5 1 1 2 3 4 5 2 2 2 6 4 10 3 3 5 3 12 15 4 4 4 12 4 20 5 5 10 15 20 5 Since,, all the elements are not in the set {1, 2, 3, 4, 5}. If we consider a = 2 and b = 3, a * b = LCM...

\[\begin{array}{*{35}{l}} a~*~b~=\text{ }2a~+~b~-\text{ }3  \\ 3\text{ }*\text{ }4\text{ }=\text{ }2\text{ }\left( 3 \right)\text{ }+\text{ }4\text{ }-\text{ }3  \\ =\text{ }6\text{ }+\text{...

(iii) Since, on R, define by a*b = ab2 Let \[\begin{array}{*{35}{l}} a,\text{ }b\text{ }\in \text{ }R  \\ \Rightarrow \text{ }a,\text{ }{{b}^{2}}~\in \text{ }R  \\ \Rightarrow \text{ }a{{b}^{2}}~\in...

(i)Since, On Z+, defined * by a * b = a – b If a = 1 and b = 2 in Z+, then \[\begin{array}{*{35}{l}} a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }-\text{ }b  \\ =\text{ }1\text{ }-\text{ }2  \\...

(vii)Since, ‘*’ on Q defined by a * b = (a – 1)/ (b + 1) for all a, b ∈ Q If a = 2 and b = -1 in Q, \[\begin{array}{*{35}{l}} a\text{ }*\text{ }b\text{ }=\text{ }\left( a\text{ }-\text{ }1...

(v) Given ‘+6’ on S = {0, 1, 2, 3, 4, 5} defined by a +6 b Consider the composition table, +6 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4 Here all...

(iii)  Given ‘*’ on N defined by a * b = a + b – 2 for all a, b ∈ N \[\begin{array}{*{35}{l}} If~a~=\text{ }1\text{ }and~b\text{ }=\text{ }1,  \\ a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }+\text{...

(i) Given ‘*’ on N defined by a * b = ab for all a, b ∈ N. Let a, b ∈ N. Then, \[\begin{array}{*{35}{l}} {{a}^{b~}}\in ~N~~~~~~\left[ \because ~{{a}^{b}}\ne 0~and~a,\text{ }b~is~positive~integer...