Chapter 9: Relations

To determine the poset \((Z, \prec )\) is well defined but is not a totally ordered set.

Find the smallest equivalence relation on the set \(\{ a,b,c,d,e\} \) containing the relation \(\{ (a,b),(a,c),(d,e)\} \).

To prove that the relation \(R\) on set \(A\) is reflexive, if and only if the complementary relation is irreflexive.

Show thata dense poset with at least two elements that are comparable is not well-founded.

To determine if the Boolean sum of two equivalence relations is an equivalence relation.

To determine the poset of rational numbers with the usual less than or equal to relation \((Q, \le )\) is a dense poset.

To provethat \({R^n} = R\forall n \in {z^ + }\)when \(R\) is reflexive and transitive.

To determine \(\left( {\frac{1}{2}} \right)\), the equivalence class of \(\frac{1}{2}\), for the given relation \(R\).

Show that the set of strings of lowercase English letters with lexigraphic order is neither well-founded nor dense.

(a) To find Relation\({R^2}\)

(b) To find Relation \({R^3}\)

(c) To find Relation \({R^4}\)

(d) To find Relation\({R^5}\)