Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
(i) On Z⁺, defined * by a * b = a − b
(ii) On Z⁺, defined * by a * b = ab
(iii) On R, define by ab = ab²
(iv) On Z⁺ define by a * b = |a − b|
(v) On Z⁺, define * by a * b = a
(vi) On R, define * by a * b = a + 4b²
Here, Z⁺ denotes the set of all non-negative integers.

Open in App

Solution

(i) If a = 1 and b = 2 in Z⁺, then

$a * b = a - b = 1 - 2 = - 1 \notin Z^{+} [∵ Z^{+} is the set of non-negative integers] \Rightarrow For a = 1 and b = 2, a * b \notin Z^{+} Thus, * is not a binary operation on Z^{+} .$

$(ii) a, b \in Z^{+} \Rightarrow a b \in Z^{+} \Rightarrow a * b \in Z^{+} Therefore, a * b \in Z^{+}, \forall a, b \in Z^{+} Thus, * is a binary operation on Z^{+} .$

$(iii) a, b \in R \Rightarrow a, b^{2} \in R \Rightarrow a b^{2} \in R \Rightarrow a * b \in R Thus, * is a binary operation on R .$

$(iv) a, b \in Z^{+} \Rightarrow |a - b| \in Z^{+} [∵ |a - b| is a positive integer] \Rightarrow a * b \in Z^{+} Therefore, a * b \in Z^{+}, \forall a, b \in Z^{+} Thus, * is a binary operation on Z^{+} .$

$(v) a, b \in Z^{+} \Rightarrow a \in Z^{+} \Rightarrow a * b \in Z^{+} Therefore, a * b \in Z^{+}, \forall a, b \in Z^{+} Thus, * is a binary operation on Z^{+} .$

$(vi) a, b \in R \Rightarrow a, 4 b^{2} \in R \Rightarrow a + 4 b^{2} \in R \Rightarrow a * b \in R Therefore, a * b \in R, \forall a, b \in R Thus, * is a binary operation on R .$

Suggest Corrections

Similar questions

Determine whether or not each of the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation, give justification for this.
(i) On $Z^{+}$ , defined $*$ by $a * b = a - b$

(ii) On $Z^{+}$ , defined $*$ by $a * b = a b$

(iii) On R, defined $*$ by $a * b = a b^{2}$

(iv) On $Z^{+}$ , defined $a * b = | a - b |$

(v) On $Z^{+}$ , defined $*$ by $a * b = a$

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

(i) On Z⁺, define * by a * b = a − b

(ii) On Z⁺, define * by a * b = ab

(iii) On R, define * by a * b = ab²

(iv) On Z⁺, define * by a * b = |a − b|

(v) On Z⁺, define * by a * b = a

Q. Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this. (i) On Z + , define * by a * b = a − b (ii) On Z + , define * by a * b = ab (iii) On R , define * by a * b = ab 2 (iv) On Z + , define * by a * b = | a − b | (v) On Z + , define * by a * b = a

Q. Determine whether or not each of the definition of

*

given below gives a binary operation. In the event that

*

is not a binary operation, given justification for this
(i) On

Z^{+}

, define

*

a * b = a - b

(ii) On

Z^{+}

, define

*

a * b = a b

(iii) On

R

, define

*

a * b = a b^{2}

(iv) On

Z^{+}

, define

*

a * b = | a - b |

Z^{+}

, define

*

a * b = a

Determine whether or not each of the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation, give justification for this.
(v) On $Z^{+}$ , defined $*$ by $a * b = a$

Join BYJU'S Learning Program