(a,b)=(c,d) if and only if a=c and b=d


Following is a proof that the ordered pairs (a,b) and (c,d) are equal if and only if a=c and b=d.

Proof.

If a=c and b=d, then (a,b)={{a},{a,b}}={{c},{c,d}}=(c,d).

Assume that (a,b)=(c,d) and a=b. Then {{c},{c,d}}=(c,d)=(a,b)={{a},{a,b}}={{a},{a,a}}={{a},{a}}={{a}}. Thus, {c,d}{{a}}. Therefore, {c,d}={a}. Hence, a=c and a=d. Since it was also assumed that a=b, it follows that a=c and b=d.

Finally, assume that (a,b)=(c,d) and ab. Then {a}{a,b}. Note that {{a},{a,b}}=(a,b)=(c,d)={{c},{c,d}}. Thus, {c}{{a},{a,b}}. It cannot be the case that {c}={a,b} (lest a=c=b). Thus, {c}={a}. Therefore, a=c. Hence, {{a},{a,b}}={{c},{c,d}}={{a},{a,d}}. Note that {a,b}{{a},{a,d}}. Since {a}{a,b}, it must be the case that {a,b}={a,d}. Thus, b{a,d}. Since ab, it must be the case that b=d. It follows that a=c and b=d. ∎

Title (a,b)=(c,d) if and only if a=c and b=d
Canonical name abcdIfAndOnlyIfAcAndBd
Date of creation 2013-03-22 16:13:19
Last modified on 2013-03-22 16:13:19
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 9
Author Wkbj79 (1863)
Entry type Proof
Classification msc 03-00