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I'm coming from a physics background, but find pure mathematics extremely interesting, so have decided to try and gain a more fundamental understanding of the subject. I've recently been reading up on relations and how one can define them as sets of ordered pairs. I am particularly interested in how one can define the equality and inequality relations of real numbers in terms of sets of ordered pairs, and from this be able to prove the transitive properties of both relations (and the symmetry and reflexive properties of equality). So far, however, I've been unable to find any notes (pdfs) that give an explanation that I can grasp conceptually. For instance, I've seen equality defined as [tex]S\subseteq\mathbb{R}^{2}=\lbrace (x,x)\in\mathbb{R}^{2} \vert\; x\in\mathbb{R}\rbrace[/tex] but I have to admit (much to my embarrassment) that I'm struggling to understand what this "means" conceptually?! Isn't one required to explicitly state the relation that two elements of [itex]\mathbb{R}[/itex] must satisfy in order to be in this set? Or is it just that one states that an element [itex]x\in\mathbb{R}[/itex] is always equal to itself and hence we can construct a set of ordered pairs containing only one element, i.e. the set containing the single element [itex]\lbrace x\rbrace[/itex] such that [tex](x,x)=\lbrace\lbrace x\rbrace, \lbrace x,x\rbrace\rbrace = \lbrace\lbrace x\rbrace\rbrace[/tex]
As such, given that the defining property of ordered pairs, [tex](a,b)=(c,d)\iff a=c \wedge b=d[/tex] it follows that [tex](x,x)=(x,y)\iff x=y[/tex] thus defining the equality relation between two elements of [itex]x\in\mathbb{R}[/itex]?!
As such, given that the defining property of ordered pairs, [tex](a,b)=(c,d)\iff a=c \wedge b=d[/tex] it follows that [tex](x,x)=(x,y)\iff x=y[/tex] thus defining the equality relation between two elements of [itex]x\in\mathbb{R}[/itex]?!