Answer
Verified
428.1k+ views
Hint: Using definition of binary operation.
(i) On ${Z^ + }$ , the binary operation$ * $ defined by $a * b = a - b$ is not a binary operation
Because if the points are taken as $\left( {1,2} \right)$ , then by applying binary operation, it becomes $1 - 2 = - 1$ and $ - 1$ does not belong to${Z^ + }$ .
(ii) On ${Z^ + }$ , the binary operation$ * $ defined by$a * b = ab$ is a binary operation because each element in ${Z^ + }$ has a unique element in ${Z^ + }$ .
(iii) On $R$ , the binary operation$ * $ defined by $a * b = a{b^2}$ is a binary operation because each element in $R$ has a unique element in $R$ .
(iv) On ${Z^ + }$ , the binary operation $ * $ defined by $a * b = \left| {a - b} \right|$ is a binary operation because each element in ${Z^ + }$ has a unique element in ${Z^ + }$ .
(v) On ${Z^ + }$ , the binary operation $ * $ defined by $a * b = a$ is a binary operation because each element in \[{Z^ + }\] has a unique element in \[{Z^ + }\] .
Note: - In order to prove that a given operation is not a binary operation just as in case I, we just need to show an example satisfying that the operation is not binary. But in all other cases, or to show that the given operation is binary we need to consider all the different possibilities and also some exceptional cases.
(i) On ${Z^ + }$ , the binary operation$ * $ defined by $a * b = a - b$ is not a binary operation
Because if the points are taken as $\left( {1,2} \right)$ , then by applying binary operation, it becomes $1 - 2 = - 1$ and $ - 1$ does not belong to${Z^ + }$ .
(ii) On ${Z^ + }$ , the binary operation$ * $ defined by$a * b = ab$ is a binary operation because each element in ${Z^ + }$ has a unique element in ${Z^ + }$ .
(iii) On $R$ , the binary operation$ * $ defined by $a * b = a{b^2}$ is a binary operation because each element in $R$ has a unique element in $R$ .
(iv) On ${Z^ + }$ , the binary operation $ * $ defined by $a * b = \left| {a - b} \right|$ is a binary operation because each element in ${Z^ + }$ has a unique element in ${Z^ + }$ .
(v) On ${Z^ + }$ , the binary operation $ * $ defined by $a * b = a$ is a binary operation because each element in \[{Z^ + }\] has a unique element in \[{Z^ + }\] .
Note: - In order to prove that a given operation is not a binary operation just as in case I, we just need to show an example satisfying that the operation is not binary. But in all other cases, or to show that the given operation is binary we need to consider all the different possibilities and also some exceptional cases.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Select the word that is correctly spelled a Twelveth class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What organs are located on the left side of your body class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE