Question

Determine whether each of the following relations are reflexive, symmetric and transitive:
(i)Relation R in the set A = {1, 2, 3...13, 14} defined as
R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as R = {(x, y): x − y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
(d) R = {(x, y): x is wife of y}
(e) R = {(x, y): x is father of y}

Answer 1

(i) (a) Given that, A={1,2,.....,13,14} and R is ={(x,y)}:3x-y =0} for reflexive $(x,x)ϵR\forall xϵA$
$\therefore$ If $y=x,3x-y=0\Rightarrow 2x\ne 0,R$ is not reflexive.
$\Rightarrow (x,x)\text{/}ϵR.$

(b)Given R ={(x,y):3x-y=0} for symmetric relation $(x,y)ϵR$
$\Rightarrow (y,x)ϵR$
If $(x,y)ϵR\Rightarrow 3x-y=0$, then $3y-x\ne 0\Rightarrow (y,x)\text{/}ϵR$, so R is not symmetric.
For example; if x =1, y =3, $3\times 1-3=0,3\times 3-1=9-1=8\ne 0$

(c)For transitive, if $(x,y)ϵR\Rightarrow 3x-y=0,(y,z)ϵR\Rightarrow 3y-z=0$, then $3x-z\ne 0$. So, R is not transitive
For example, when x=1, y=3, z=9, then $3\times 1-3=0,3\times 3-9=0,3\times 1-9\ne 0$

Here, A=N, the set of natural numbers and R={(x,y);y=x+5,x<4}
={(x,x+5):$xϵN$ and x < 4}

={(1,6),(2,7),(3,8)}
(a)For reflexive $x,xϵR\forall x$ putting $y=x,x\ne y+5\Rightarrow (1,1)\text{/}ϵR$. So, R is not reflexive.

(b)For symmetrical $(x,y)ϵR\Rightarrow (y,x)ϵR$ putting y =x+5, then $x\ne y+5\Rightarrow (1,6)ϵR$ but $(6,1)\text{/}ϵR$. So, R is not symmetric.

(c)For transitivity $(x,y)ϵR,(y,z)ϵR\Rightarrow (x,z)ϵR$ if y =x+5, z=y+5, then $z\ne x+5$. Since, $(1,6)ϵR$ and there is no order pair in R which has 6 as the first element same in the case for (2,7)and (3,8). So, R is not transitive

(a) For reflexive, we know that x is divisible by x for all $xϵA$.
$\therefore (x,x)ϵR$ for all $xϵR$. So, R is reflexive.

(b)For symmetry, we know that x is divisible by 2. This means that $(2,6)ϵR$ but $(6,2)doesnotbelongtoR$. So, R is not symmetric.

(c)For transitiviety, let (x,y)\ belongs R and $(y,z)ϵR$, then z is divisible by x.
$\Rightarrow (x,z)ϵR$
For example, 2 is divisible by 1,4 is divisible by 2
So, 4 is divisible by 1 so, R is transitive.

For reflexive put y=x, x-x =0 which is an integer for all $xϵZ$. So, R is reflexive on Z.
(b)For symmetry let $(x,y)ϵR$, then (x-y)is an integer $\lambda$ and also $y-x=-\lambda [\because \lambda ϵZ\Rightarrow -\lambda ϵZ]$
$\therefore y-x$ is an integer $\Rightarrow (y,x)ϵR$. SO, R is symmetric.
(c)For transitivity let (x,y) belongs R and $(y,z)ϵR\therefore x-y$ = integer and y-z = integer, then x-z is also an integer
$\therefore (x,z)ϵR$. So, R is transitive

(a)R is reflexive, symmetric and transitive obviously.( For reflexive (x, x ) work in same place. For symmetric (x,y) and (y,x) work in same place. For transitive (x,y) , (y,z) work at same place place means (x,z) work at same place)
(b)R is reflexive, symmetric and transitive obviously.( For reflexive (x, x ) live in same locality. For symmetric (x,y) and (y,x) live in same locality. For transitive (x,y) , (y,z) live in same locality means (x,z) live in same locality)
(c)Here, R is not reflexive as x is not 7 cm taller than x itself . R is not symmetric as if x is exactly 7 cm taller than y, then y exactly 7 cm taller than x is incorrect and not transitive as x is exactly 7 cm taller than y and y is exactly 7 cm taller than z , then x is exactly 7 cm taller than z is incorrect
(d)Here, R is not reflexive; as x is not wife of x, R is not symmetric, as if x is wife of y. then y is husband (not wife)of x and R is not transitive as x is wife of y and y is wife of z is contradicted .
(e)Here, R is not reflexive; as x cannot be father of x. For any x, R is not symmetric as if x is father of y, then y cannot be father of x. R is not transitive as if x is father of y and y is father of z, then x is grandfather (not father)of z.