How to represent binary relations?

Question

Basically, Im not sure how to to evaluate if a ∧ b is a function, injective, reflexive etc. I can evaluate these properties if Its a real function e.g. f(x) = x+2 as I can just look at the graph/test these properties using numbers. But I have no idea how to represent "a ∧ b" as a graph/in a relationship diagram. To reiterate, I understand the definitions of injective, reflexive etc, I just dont know how to apply them to binary relations.

Answer 1

Try $\;\;\text{graph} \;(C) = \{(a, b) \in \mathbb Z \times \mathbb Z\mid C(a, b) = T\},\;$ where the binary relation is $"\land$" $\;$ and so $aCb \iff a\land b = T$

$ (T\land T) = T, \quad (T \land F) = F, \quad (F\land T) = F$, $ (F\land F) = F $

So $C = \{(T,T)\}$

Note that the relation $\text{graph}C = \{(a, b)= (T, T)\}$, (because $a\land b$ is true only when $a =b= T$. Clearly, Since we need a = b = T, $a C a =T$, and $b C b = T$. So the relation is reflexive. It is also symmetric, since $(a, b) = (T, T) = (b, a)$

Answer 2

I do not see the problem. You can view $\wedge$ as function $$ \wedge : \{0,1\}^2 \to \{0,1\} $$ and verify it is a function (it is single valued at each point). It is not injective, as $$ \wedge(t,f) = \wedge(f,t) = f $$ so several arguments point to the same value.

To see if it is reflexive, you have to view it as relation: $$ \wedge(t,t) = t \iff t \wedge t \\ \wedge(f,f) = f \iff \neg (f \wedge f) $$ so it is not reflexive. Etc. A table could be: $$ \begin{array}{r|rr} x \wedge y & f & t \\ \hline f & - & - \\ t & - & \surd \end{array} $$