# Each of the following defines a relation on $N : (iii)\quad x\;y\;is\;square\; of\; an\; integer\;x,y\quad N$

Determine which of the above relation is

A) reflexive

B) symmetric

C) transitive.

Note: This is the 3rd part of a  4 part question, which is split as 4 separate questions here.

Toolbox:
• 1. A function R defined on A reflexive if $(x,x) \in R\;for\;x \in A$
• 2.A relation R defined on A is symmetric if $(x,y) \in R =>(y,x) \in R \qquad x,y \in A$
• 3. A relation R defined on A is transitive if $(x,y) \in R (y,z) \in R =>(x_1,x_2) \in R. x,y,z \in A$
(iii) $R=\{(x,y):xy\; is\; square\; of\; integer\; x,y\; \in N\}$

Consider $(x,x) \in R$ since

$x \times x =x^2=$square of integer

Therefore R is reflexive

$(x,y) \in R$

$=>x \times y =$ square of integer

$y \times x=$square of integer

Multiplication is commutative in N

$=>(y,x) \in R$

Therefore R is symmetric

$(x,y) \in R ; (y,z) \in R$

xy=square of integer

yz=square of integer

$xz=\frac{(xy)(yz)}{y^2}$

Since xz is product of square of integers

xz=square of integer

$(x,z)\in R$

R is transitive

R is reflexive,symmetric and transitive