244 Logic as a Tool and explicating the logical structure of the argument Distinguish the logical steps from the mathematical steps 5.2.10 Prove Proposition 192 5.2.13 Prove Proposition 195 5.2.11 Prove Proposition 193 5.2.14∗ Prove Proposition 196 5.2.12 Prove Proposition 194 Exercises on binary relations For each of the following exercises consider a suitable first-order language extending LZF with the relational symbols mentioned in the exercise Then formalize the respective property in that language and attempt to prove it using a deductive system of your choice Provide semi-formal mathematical proofs, as required, by indicating the method of proof and explicating the logical structure of the argument Distinguish the logical from the mathematical steps 5.2.15 Prove Proposition 197 5.2.18 Prove Proposition 200 5.2.16 Prove Proposition 198 5.2.19 Prove Proposition 202 5.2.17 Prove Proposition 199 5.2.20 Show that a relation R ⊆ A × B is functional iff it is the graph of a function from A to B 5.2.21 Show that each of the reflexive, symmetric, and transitive closures of any given binary relation exists 5.2.22 Prove Proposition 203 5.2.23∗ Prove each of the following for any binary relations R, S, T ⊆ X (a) If R and S are transitive and R ◦ S = S ◦ R = T , then T is also transitive (b) If R and S are transitive, R is symmetric, and R ∪ S = X , then R = X or S = X (c) If R or S is reflexive and transitive and R ◦ S = EX , then R = S = EX 5.2.24 Prove Proposition 204 5.2.26 Prove Proposition 206 5.2.25 Prove Proposition 205 5.2.27 Prove Proposition 207 5.2.28 Prove that R ⊆ X is an equivalence relation iff R is reflexive and euclidean 5.2.29∗ Let E1 and E2 be equivalence relations on a set X Then prove that: (a) E1 ∩ E2 is an equivalence relation on X (b∗) The composition E1 ◦ E2 is an equivalence relation if and only if the two relations commute, that is, if and only if E1 ◦ E2 = E2 ◦ E1 5.2.30 Prove that if (X, ≤) is a poset and Y ⊆ X , then (Y, ≤|Y ) is also a poset