Question

The defining property of an ordered pair is that two ordered pairs are equal if and only if their first 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 \(\left( {{\bf{a,b}}} \right)\)to be\(\left\{ {\left\{ {\bf{a}} \right\}{\bf{,}}\left\{ {{\bf{a,b}}} \right\}} \right\}\), then \(\left( {{\bf{a,b}}} \right){\bf{ = }}\left( {{\bf{c,d}}} \right)\)

(Hint: First show that \(\left( {\left\{ {\bf{a}} \right\}{\bf{,}}\left\{ {{\bf{a,b}}} \right\}} \right){\bf{ = }}\left( {\left\{ {\bf{c}} \right\}{\bf{,}}\left\{ {{\bf{c,d}}} \right\}} \right)\)if and only if \({\bf{a = c}}\)and

\({\bf{b = d}}\)).

Short answer

\(\left\{ {a,b} \right\} = \left\{ {c,d} \right\}\)if \(a = c\) and \(b = d\)

Step-by-step solution

To use the given statement

\(\left( {\left\{ {\bf{a}} \right\}{\bf{,}}\left\{ {{\bf{a,b}}} \right\}} \right){\bf{ = }}\left( {\left\{ {\bf{c}} \right\}{\bf{,}}\left\{ {{\bf{c,d}}} \right\}} \right)\)if and only if \({\bf{a = c}}\)and \({\bf{b = d}}\)

To show that\(\left\{ {{\bf{a,b}}} \right\}{\bf{ = }}\left\{ {{\bf{c,d}}} \right\}\)if\({\bf{a = c}}\)and \({\bf{b = d}}\)

Here, it is given that:

\(\left( {a,b} \right) = \left\{ {\left\{ a \right\},\left\{ {a,b} \right\}} \right\}\)

To Prove: \(\left( {a,b} \right) = \left( {c,d} \right)\)

First part:

Assume \(\left( {a,b} \right) = \left( {c,d} \right)\)

By definition of ordered pairs:

\(\left( {a,b} \right) = \left\{ {\left\{ a \right\},\left\{ {a,b} \right\}} \right\}\)

Also,\(\left\{ {\left\{ a \right\},\left\{ {a,b} \right\}} \right\} = \left\{ {\left\{ c \right\},\left\{ {c,d} \right\}} \right\}\)

Two ordered pairs are same if the corresponding elements are same.

\(\left\{ a \right\} = \left\{ c \right\}\)

\(\left\{ {a,b} \right\} = \left\{ {c,d} \right\}\)

Since, \(\left\{ a \right\} = \left\{ c \right\}\)

Therefore, \(a = c\)

This implies that \(\left\{ {c,d} \right\} = \left\{ {a,b} \right\}\)

The two sets that contain 2 elements each and have one element in common are equal when the second element in the two sets are identical.

\(\left\{ {a,b} \right\} = \left\{ {c,d} \right\} = \left\{ {a,d} \right\}\)

Therefore, \(b = d\)

Thus, it is shown that \(a = c\) and \(b = d\)

Second part:

Assume \(a = c\) and \(b = d\)

By definition of ordered pairs:

\(\begin{array}{c}\left( {a,b} \right) = \left\{ {\left\{ a \right\},\left\{ {a,b} \right\}} \right\}\\ = \left\{ {\left\{ c \right\},\left\{ {c,b} \right\}} \right\}\\ = \left\{ {\left\{ c \right\},\left\{ {c,d} \right\}} \right\}\\ = \left\{ {c,d} \right\}\end{array}\)

Thus, it is shown that \(\left\{ {a,b} \right\} = \left\{ {c,d} \right\}\).