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Each and every function $f:X\rightarrow Y$ between two totally ordered sets $(X,\leq_X), (Y,\leq_Y)$ induces a relation $\preceq$ on $X$ by $$x \preceq x' :\equiv \begin{cases} x &\leq_X x' ... 1answer 109 views ### Deriving the Axiom of Infinity I'm currently reading about recursion, working on various exercises since I haven't formally studied the material before. The main book that I'm following is Kunen's Set Theory book There's an ... 1answer 70 views ### Operations and relations To what extent do operations and relations overlap? Is there some more general structure that encompasses both of these things? Thanks 1answer 134 views ### A characteristic of intersection with cartesian product Fix some binary relation f. Does there necessarily exist a set C such that (x\times x)\cap f\ne \varnothing \Leftrightarrow x\cap C\ne \varnothing for all sets x? 1answer 138 views ### < on a preorder is a strict partial order Definition: Suppose X is a preorder. Define x < y as x \le y and y \not\le x for each x, y \in X. Question: Show that this gives a strict partial order on X. 1answer 209 views ### Preorders, chains, cartesian products, and lexicographical order Definitions: A preorder on a set X is a binary relation \leq on X which is reflexive and transitive. A preordered set (X, \leq) is a set equipped with a preorder.... Where confusion cannot ... 2answers 134 views ### < in a preorder The author of the book I am studying defines < for a poset as If x, y \in X, where X is a poset, then we shall write x < y to mean that x \le y and x \ne y. From this, I can ... 1answer 206 views ### Order of products and order of multipliers I asked this question (and have received an answer) at MathOverflow. Now a little more difficult question: Let f and g are binary relations (on some set \mho). The function f\times^{C} g is ... 1answer 129 views ### Find an elegant proof of a set-theoretic equiality about relations I am now attempting to prove the following theorem. I am in half-underway of the proof and it seems I can do it by myself. But the proof I am now constructing is not elegant. Could anyone provide a ... 2answers 269 views ### Unnecessary property in definition of equivalence relation [duplicate] Possible Duplicates: Symmetric, Transitive and reflexive Why isn't reflexivity redundant in the definition of equivalence relation? Dependence of Axioms of Equivalence Relation? Let ... 1answer 452 views ### A counter-example for a set-theoretic problem? I have proved the below conjecture for the special cases n\in\{0,1,2\}. The cases n\ge 3 (finite and infinite) are unknown. If the following conjecture is true, I don't expect that you will be ... 1answer 95 views ### Simplify a formula about relations Let F is an n-ary relation (with n being any index set). Can the following formula be simplified?$$(\lambda x\in n:s(x))\in F ($s$ is some function). Here $\lambda$ is defined as: ...
Working in ZFC. I've defined a function-like binary predicate $R$ on a proper class. It has to be recursive; i.e. $R(a,b)$ must usually depend on one or more $R(c,d)$ for some $c$s and $d$s ...
A relation $\mathrm{R}$ is defined on the set of all positive integers by: $x\mathrm{R}y$ if and only if $y = 3^k\cdot x$ for some non-negative integer $k$. Prove that $\mathrm{R}$ is a partial ...