Question

a) Provide an example of a relation T on set A = {1, c, 3} that is irreflexive and satisfies trichotomy. (What is Trichotomy)

b) Let S= {(1, 2), (3, 4), (2, 3)} be a relation on set B = {1, 2, 3, 4}.
Is S functional? Motivate your answer

c) Let C = {1, 2, a, b} and let R = {(1, 1), (a, b), (b, 2)} and S = {(2, 1), (a, 1), (b, b), (b, 2), (2, a)} be two relations on C
i) Determine R o S (S;R)
ii) Which ordered pairs must be added to R to make it a reflexive relation?

Answer 1

a) On the set $X = \{1,c,3\}$, the relation $R = \{ (1,c), (1,3), (c,3)\}$ is irreflexive because $\forall x\in X\colon (x,x)\notin R$, and trichotomous (see first example in [1], it is the same). $R$ is trichotomous (see first property in [1]) because it is asymmetric, $(x,y)\in R \rightarrow (y,x)\not\in R$, and semiconnex, $\forall x \ne y\in X\colon (x,y)\in R \, \lor \, (y,x)\in R$.

b) Yes, because it is one-to-one

c) i) $S;R$ means that firstly $S$ then $R$. Hence

$R\circ S=\{(2,1),(a,1),(b,2),(2,b)\}$

ii) $(1,1), (2,2), (a,a), (b,b)$

[1] https://en.wikipedia.org/wiki/Trichotomy_(mathematics)