Applications: Mathematical Proofs and Automated Reasoning 241 Proposition 207 Let R be a pre-order on X Then: (a) The relation ∼ on X defined by x ∼ y iff xRy and yRx is an equivalence relation on X ˜ on X/ , defined by (b) The relation R ∼ ˜ [y ] iff xRy [x]∼ R ∼ is a well-defined partial order on X/∼ , called the partial order induced by R 5.2.6.2 Lower and upper bounds; minimal and maximal elements Let (X, ≤) be a poset and Y ⊆ X An element x ∈ X is: • a lower bound for Y in X if x ≤ y for every y ∈ Y ; • the greatest lower bound, also called infimum, of Y in X if x is a lower bound for Y in X and x ≤ x for every lower bound x of Y in X ; • an upper bound for Y in X if x ≥ y for every y ∈ Y ; or • the least upper bound, also called supremum, of Y in X if x is an upper bound for Y in X and x ≤ x for every upper bound x of Y in X Proposition 208 Let (X, ≤) be a poset and Y ⊆ X (a) If Y has an infimum in X , then it is unique (b) If Y has a supremum in X , then it is unique Let (X, ≤) be a poset and Y ⊆ X An element x ∈ Y is called: • minimal in Y if there is no element of Y strictly less than x, that is, for every y ∈ Y , if y ≤ x then x = y ; or • maximal in Y if there is no element of Y strictly greater than x, that is, for every y ∈ Y , if y ≥ x then x = y Note that the least (respectively, greatest) element of Y , if it exists, is the only minimal (respectively, maximal) element of Y 5.2.6.3 Well-ordered sets An ascending (respectively, strictly ascending) chain in a poset (X, ≤) is any finite or infinite sequence x1 , x2 , x3 , of elements of X such that x1 ≤ x2 ≤ x3 ≤ · · · (respectively, x1 < x2 < · · ·) A descending (respectively, strictly descending) chain in a poset