What is the Kuratowski Definition of an Ordered Pair in Set Notation?

[Gear300]

The Kuratowski definition of the ordered pair is (a,b) = {{a},{a,b}}...this sort of lost me...how did they define an ordered pair (in which order of elements matters) using set notation (how does this definition work)?

 

[g_edgar]

An ordered pair (a,b) is supposed to be some object with this property:

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

What it actually *is* we don't care, as long as it has this property. Kuratowski wrote down his definition as something that has this property.

 

[Gear300]

An ordered pair (a,b) is supposed to be some object with this property:

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

What it actually *is* we don't care, as long as it has this property. Kuratowski wrote down his definition as something that has this property.

I see...I thought they were defining (a,b) with the expression. So, in that sense, the expression does not show that order matters, right?

 

[Werg22]

I see...I thought they were defining (a,b) with the expression. So, in that sense, the expression does not show that order matters, right?

Yes it does. Notice that if a is not equal to b, the RHS becomes {{b}, {b,a}} which is definitely not equal to {{a}, {a,b}}. Also, if {{a}, {a,b}} = {{x}, {x,y}}, then necessarily a = x and b = y.

This is just a way of capturing the notion of ordered pairs with a set-theoretic definition. What ordered pairs actually are, like g_edgar said, doesn't really matter: as long your notion/definition of them has the desired properties, then there is no harm done.

 

[Gear300]

I see...Thanks for the replies...but isn't {{a},{a,b}} just another way of writing {a,b}?

 

[Werg22]

I see...Thanks for the replies...but isn't {{a},{a,b}} just another way of writing {a,b}?

No... {a,b} is not the same as {{a}, {a,b}}. The members of the first set are a and b, the members of the second are {a} and {a,b}. The members of the second set are NOT a and b, if this is what you are implying.

 

[HallsofIvy]

I see...Thanks for the replies...but isn't {{a},{a,b}} just another way of writing {a,b}?

{1,3} is the same as {3,1}. But {{1},{1,3}} is not the same as {{3},{3,1}}. Kuratowski's definition basically states that there are two elements and distinguishes between the two, thus giving an order.

 

[Gear300]

Thus, the elements are different...I think I see what's going on to a better extent. Thanks for the replies.