(iii) Since, on R, define by a*b = ab2
Let
![\[\begin{array}{*{35}{l}} a,\text{ }b\text{ }\in \text{ }R \\ \Rightarrow \text{ }a,\text{ }{{b}^{2}}~\in \text{ }R \\ \Rightarrow \text{ }a{{b}^{2}}~\in \text{ }R \\ \Rightarrow \text{ }a\text{ }*\text{ }b\text{ }\in \text{ }R \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bd99cbb540f562eb785d7cea485bd4a6_l3.png "Rendered by QuickLaTeX.com")
Thus, * is a binary operation on R.
(iv) Given on Z+ define * by a * b = |a − b|
Let
![\[\begin{array}{*{35}{l}} a,\text{ }b\text{ }\in \text{ }{{Z}^{+}} \\ \Rightarrow \text{ }\left| \text{ }a\text{ }-\text{ }b\text{ } \right|\text{ }\in \text{ }{{Z}^{+}} \\ \Rightarrow \text{ }a\text{ }*\text{ }b\text{ }\in \text{ }{{Z}^{+}} \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-77398e6ffb80eda8661328162f35e9d5_l3.png "Rendered by QuickLaTeX.com")
Therefore,
a * b ∈ Z+, ∀ a, b ∈ Z+
Thus, * is a binary operation on Z+.
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.(iii) On R, define * by a*b = ab2 (iv) On Z+ define * by a * b = |a − b|
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.(iii) On R, define * by a*b = ab2 (iv) On Z+ define * by a * b = |a − b|