Determine whether or not the definition of * | vedclass

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(B) On the set of positive integers $Z^{+}$,the operation $*$ is defined by $a * b = a - b$. $A$ binary operation $*$ on a set $S$ is a function $*: S

Vedclass · Accepted Answer

No,it is not a binary operation because $a - b$ is not always in $Z^{+}$.

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(B) On the set of positive integers $Z^{+}$,the operation $*$ is defined by $a * b = a - b$. $A$ binary operation $*$ on a set $S$ is a function $*: S 	imes S 	o S$. For the operation to be binary,for every pair $(a, b) \in Z^{+} 	imes Z^{+}$,the result $a * b$ must belong to $Z^{+}$. Consider the elements $a = 1$ and $b = 2$,where $1, 2 \in Z^{+}$. Then $a * b = 1 * 2 = 1 - 2 = -1$. Since $-1 
otin Z^{+}$,the operation $*$ is not 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 for this. On $Z^{+}$,define $$ by $a b = a - b$.

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