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Question

1) Mentioned below defines a relation on N:

(i) x is greater than y,x,yN

Determine whether the relation is reflexive, symmetric and transitive.

2) Mentioned below defines a relation on N:

(ii) x+y=10,
x,yN

Determine whether the relation is reflexive, symmetric and transitive.

3) Mentioned below defines a relation on N:

(iii) xy is square of an integer x,yN

Determine whether the relation is reflexive, symmetric and transitive.

4) Mentioned below defines a relation on N:

(iv) x+4y=10,x,yN

Determine whether the relation is reflexive, symmetric and transitive.


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Solution

1) Testing Reflexivity

Given relation on N:

x is greater than y,x,y N

If (x,x)ϵR, then x>x, which is not true for any xN

(x,x)R

So, R is not reflexive.

Testing whether a relation is symmetric

Let (x,y)R

x>y

y<x not possible

(y,x)R

Hence, R is not symmetric.

Testing whether a relation is transitive

Let (x,y)R and (y,z)R

x>y and y>z

x>z

(x,z)R

Hence, R is transitive.

Final Answer:R is only transitive.

2) Testing whether relation is reflexive.

Given relation on N:

x+y=10,x,yN

If (x,x)R, then x+x=10,xN

Since, 1+110

Therefore, (1,1)R

Hence, R is not reflexive

Testing whether relation is symmetric

Let (x,y)R

x+y=10

y+x=10

(a+b=b+a,a,bN)

(y,x)R

Therefore, R is symmetric

Testing whether relation is transitive

Since, x+y=10,x,yN

(1,9)R and (9,1)R

But (1,1)R, as 1+110

Therefore, R is not transitive.

Hence, R is only symmetric.

3) Checking whether the relation is reflexive

Given:

R={(x,y):xy is a square of an integerx,yN}

As, x2 is square of an integer for any xN

Therefore, (x,x)R,xN

Hence, R is reflexive

Checking whether the relation is symmetric

Let (x,y)R

xy is a square of an integer

yx is a square of an integer

( multiplication is commutative in N)

(y,x)R

Hnece, R is symmetric

Checking whether the relation is transitive

Let (x,y)R and (y,z)R

xy is square of an integer and

yz is square of an integer

Let xy=m2 and yz=n2, for some m,nZ

Thenx=m2y and z=n2y

xz=m2n2y2

which is square of an integer

(x,z)R

Hence R is transitive

Hence, R is reflexive, symmetric and transitive.

4) Writing the relation R is roster form

Given:

R:{(x,y):x+4y=10,x,yN}

In roster form: R={(6,1),(2,2)}

Clearly, (1,1)R

Therefore, R is not reflexive

Now, Since, (6,1)R, but (1,6)R

Therefore, R is not symmetric

Since, there are no elements of the form (a1,a2),(a2,a3) in the given relation.

Hence, R is transitive.

R R is only transitive and it is neither reflexive nor symmetric.


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