Answer : D
Let $A$ be Set of All functions from $Z$ to $Z$.
Option A :
$R_1$ = $\left \{ (f,g) | f(x) = g(x) = 1, \forall x \in Z \right \}$ .. It is Not Reflexive because $A$ is Set of All the functions from $Z$ to $Z$. And the given condition $f(x) = g(x) = 1, \forall x \in Z $ will only be satisfied by an Unique pair $(f,f)$. So, $R_1$ is Symmetric, Transitive But Not Reflexive.
$R_1$ = Not Reflexive, Symmetric, Transitive, Not Irreflexive, Antisymmetric, Not Asymmetric, Not an Equivalence Relation, Not a Partial Order relation.
Option B :
$R_2$ = $\left \{ (f,g) | f(0) = g(0) \,\,or\,\,f(1) = g(1) \right \}$
It is Reflexive, Symmetric But Not Transitive.
for e.g. : Let $(f,g), (g,h) \in R_2$ such that $f(0) = g(0) = 1$ and $g(1) = h(1) = 2$ and $h(0) = 4, f(1) = 5$
So, We can see that $(f,h) \notin R_2 $. Hence, Not Transitive.
$R_2$ = Reflexive, Symmetric, Not Transitive, Not Irreflexive, Not Antisymmetric, Not Asymmetric, Not an Equivalence Relation, Not a Partial Order relation.
Option 3 :
$R_3$ = $\left \{ (f,g) | f(0) = g(1) \,\,and\,\,f(1) = g(0) \right \}$
It is Not Reflexive because $A$ is Set of All the functions from $Z$ to $Z$. And the given condition $f(0) = g(1) $ and $f(1) = g(0)$ will Not be satisfied by all the Pairs $(f,f)$.
for e.g. : Let $h$ be a function such that $h(0) = 1$ and $h(1) = 2$ , So, we can see that $(h,h) \notin R_3$
$R_3$ = Not Reflexive, Symmetric, Not Transitive, Not Irreflexive, Not Antisymmetric, Not Asymmetric, Not an Equivalence Relation, Not a Partial Order relation.
See, $R_3$ is Not Transitive.
for e.g. : Let $(f,g), (g,h) \in R_3$ such that $f(0) = g(1) = 5$ and $f(1) = g(0) = 6$..So, $g(0) = h(1) = 6$ and $g(1) = h(0) = 5$
So, We can see that $(f,h) \notin R_3 $ because $f(0) = 5 $ But $h(1) = 6$ Hence, Not Transitive.
Option D :
$R_4$ = $\left \{ (f,g) | f(x) - g(x) = K, \,\,for\,\,some\,\,K \in Z \right \}$
$R_4$ is Reflexive, Symmetric, Transitive. Hence, An Equivalence Relation.
NOTE that $R_4$ is Reflexive, Symmetric, Transitive because Range of All the functions is $Z$ and We know that $Z - Z = Z$
$R_4$ = Reflexive, Symmetric, Transitive, Not Irreflexive, Not Antisymmetric, Not Asymmetric, An Equivalence Relation, Not a Partial Order relation.