Definition 3.5.1 (Ordered pair). If \(x\) and \(y\) are any objects (possibly equal), we define the ordered pair \((x,y)\) to be a new object, consisting of \(x\) as its first component and \(y\) as its second component. Two ordered pairs \((x,y)\) and \((x',y')\) are considered equal if and only if both of their components match, i.e $$ (x,y)=(x',y')\iff(x=x'\text{ and } y=y') $$