# Thread: Ordered pair. need help in understanding.

1. ## Ordered pair. need help in understanding.

Need help in understanding this theorem..

Theorem 2.1.1. Let <a, b> and <c, d> be ordered pairs. Then <a, b> = <c, d> if and only if a = c and b = d.
Remark. The expression “if and only if” means that
1. If <a, b> = <c, d> , then a = c and b = d.
2. If a = c and b = d, then <a, b> = <c, d> . (This is called "the converse" of 1.)
So, we have to prove two directions, namely 1. and 2. Usually, “if and only if” is abbreviated as simply “iff”.
Proof. “⇒”: Suppose <a, b> = <c, d> ; then, by deﬁnition of ordered pair,
<a, b> = {{a}, {a, b}},
<c, d> = {{c}, {c, d}},
and since they are equal by our hypothesis, we have
{{a}, {a, b}} = {{c}, {c, d}}.
We consider two cases:
1. a = b: Then,
<a, b> = {{a}, {a, b}} = {{a}} = {{c}, {c, d}},
hence, {a} = {c} which implies a = c. Furthermore, {a} = {c, d} = {a, d}
which implies d = a = b. Thus, for this case we have shown that a = c and b = d.

I got stuck at this d=a=b. how?

2. Originally Posted by chakravarthiponmudi
Need help in understanding this theorem..

We consider two cases:
1. a = b: Then,
<a, b> = {{a}, {a, b}} = {{a}} = {{c}, {c, d}},
hence, {a} = {c} which implies a = c. Furthermore, {a} = {c, d} = {a, d}
which implies d = a = b. Thus, for this case we have shown that a = c and b = d.

I got stuck at this d=a=b. how?
Well if you are good through {a} = {c, d} = {a, d} you are golden. the set {a} has 1 element, the other two have 2 elements, for finite sets to be equal, the cardinality must be the same. but that means c,d and a,d are really they same thing and they must all actually be a to have the sets be equal. So that gives us a=c=d, but recall which case we are in, for a=b. so we get b=a=c=d, which means that in particular a=c and b=d, they are just all the same too. It is kind of like the trivial case I guess. Does that explain it?

3. ## Thank you

Yes. I got it now.. Thankyou.