Question

Let R be a binary relation on N × N defined by (w, x ) R (y, z) if and only if w = y and x ≤ z . Is R reflexive? Is R symmetric? Is R antisymmetric? Is R transitive?

Answer 1

relation R on the set A is called reflexive if $\forall a\isin A:aRa$

$\forall n \in N:n=n$ and $\forall m \in N:m≤m$ , then $\forall (n,m) \in NXN: (n,m)R(n,m)\implies$ R is reflexive

relation R on the set A is called reflexive if $\forall a,b\isin A:aRb\implies bRa$

$\forall n,m \in N:n=m\implies m=n$ and $\forall k,t \in N:k≤t \implies t≤k$ , then $\forall n,m, k, t \in N: (n,m)R(k,t)\implies (k,t)R(n,m)\implies$ R is symmetric

relation R on the set A is called antisymmetric if $\forall a,b\isin A:(aRb\land bRa)\implies a=b$

$\forall k,t \in N:(k≤t)\land (t≤k) \implies k=t$ , then $\forall n,m, k, t \in N: ((n,m)R(k,t))\land ((k,t)R(n,m))\implies (n,m)=(k,t)\implies$ R is antisymmetric

relation R on the set A is called transitive if $\forall a,b,c\isin A:(aRb\land bRc)\implies aRc$

$\forall n,m,t \in N:(n≤m)\land (m≤t) \implies k=m$ and $\forall s, q, v \in N:(s=q)\land (q=v) \implies s=v$ , then $\forall n,m, k, t, s, v \in N: ((n,m)R(k,t))\land ((k,t)R(s,v))\implies (n,m)R(s,v) \implies$ R is transitive