The defining property of an ordered pair is that two ordered pairs are equal if and only if their first elements are equal and their second elements are equal. Surprisingly, instead of taking the ordered pair as a primitive concept, we can construct ordered pairs using basic notions from set theory. Show that if we define the ordered pair (a, b) to be {{a}, (a. b}}. then (a, b) = (c, d) if and only if a = c and b = d. [Hint: First show that {{a}. {a. b}} = {{c}, {c, d} if and only if a = c and b = d.)

SolutionStep 1Let us assume that a = c and b = d then {{a} , {a , b}} = {{c} , {c , d}}Now we have two cases in first case we assume that a = bSubstitute the value back in previous equation {{a} , {a , b}} = {{a} , {a , a}} = {{a}} {{c} , {c , d}} = {{a}}Here {c} = {c, d} = {a} which means that a = c and a = dNow, By Hypothesis we can say that a = c and b = dStep 2Next, in second case, we assume that a b {{a} , {a , b}} = {{c} , {c , d}}So, let us assume that {c ,...