In elementary mathematics, the notion of an ordered pair introduced at the begin

Question

In elementary mathematics, the notion of an ordered pair introduced at the beginning of this section will suffice. However, if we are interested in a formal development of the Cartesian product of two sets, we need a more precise definition of ordered pair. Following is one way to do this in terms of sets. This definition is credited to Kazimierz Kuratowski (1896 - 1980). Kuratowski was a famous Polish mathematician whose main work was in the areas of topology and set theory. He was appointed the Director of the Polish Academy of Sciences and served in that position for 19 years. Let x be an element of the set A, and let y be an element of the set B. The ordered pair (x, y) is defined to be the set {{x}, {x, y}}. That is, (x, y) = {{x}, {x, y}}. (a) Explain how this definition allows us to distinguish between the ordered pairs (3, 5) and (5, 3). (b) Let A and B be sets and let a, c belongs to A and b, d belongs to B. Use this definition of an ordered pair and the concept of set equality to prove that (a, b) = (c, d) if and only if a = c and b = d. An ordered triple can be thought of as a single triple of objects, denoted by (a, b, c), with an implied order. This means that in order for two ordered triples to be equal, they must contain exactly the same objects in the same order. Thai is, (a, b, c) = (p, q, r) if and only if a = p, b = q and c = r. (c) Let A, B, and C be sets, and let x belongs to A, y belongs to B, and z belongs to C. Write a set theoretic definition of the ordered triple (x, y, z) similar to the set theoretic definition of "ordered pair."

Explanation / Answer

(a) if (x,y) were simply {x,y} there would be no difference between (x,y) and (y,x) since

{x,y} = {y,x}

Since (x,y) = {x,{x,y}}, (y,x) = {y, {y,x}} and the two clearly different

(3,5) = {3,{3,5}} and (5,3) = {5,{5,3}}

The two sets have different elements and are not the same.

(b) If a=c and b=d

then (a,b) = {a,{a,b}}

={c,{c,d}}

= (c,d)

For the only if part lets assume a and c are different but b and d are same. Then

(a,b) = {a,{a,b}} = {a,{a,d}}

(c,d) = {c,{c,d}}

which are different. By a similar analogy this is true when b and d are different too.

(c) An order triple can be defined by similar definition as follows

(x,y,z) = {x,{x,(y,z)}}

Now using our earlier definition, we get

(x,y,z) = {x,{x,{y,{y,z}}}}

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