Question

Relations and Functions
Binary Operations

Do answer the question in the given pic properly
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Answer 1

Answer:

option (a) is the right answer (1,1)

Answer 2

Answer:

${\huge{\pink{↬}}} \:  \: {\huge{\underline{\boxed{\bf{\pink{Answer}}}}}}$

Consider the problem

A=N×N

(a,b)∗(c,d)=(a+b,b+d)

For commutative

Let (a,b)∈N×Nand(c,d)∈N×N

then, 

(a,b)∗(c,d)=(a+b,b+d)...(1)

And

(c,d)∗(a,b)=(c+a,d+b)=(a+c,b+d)...(ii)

From (i) & (ii)

(a,b)∗(c,d)=(c,d)∗(a,b)

So, ∗ is commutative 

For Associative 

Let (a,b),(c,d)&(e,f) belongs to A

{(a,b)∗(c,d)}∗(e,f)=(a+c,b+d)∗(e,f)=(a+c+e,b+d+f)....(4)

Also

(a,b)∗{(c,d)∗(e,f)}=(a,b)∗(c+e,d+f)=(a+c+e,b+d+f)....(5)

From (4) and (5)

{(a,b)∗(c,d)}∗(e,f)=(a,b)∗{(c,d)∗(e,f)}

So, ∗ is Associate

And Identity element does not exists.