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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.

1 Answer

  • 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
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



answered Mar 4, 2013 by meena.p

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