The discussion revolves around the Kuratowski definition of an ordered pair in set notation, specifically the expression (a,b) = {{a},{a,b}}. Participants explore how this definition captures the concept of order in pairs and the implications of this representation in set theory.
Participants generally agree on the property that defines ordered pairs but exhibit disagreement regarding the implications of the set notation and whether it adequately demonstrates the importance of order.
Some limitations in understanding arise from the nuances of set membership and the definitions involved, as well as the potential confusion between different representations of sets.
g_edgar said: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 said: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?
Gear300 said: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 said:I see...Thanks for the replies...but isn't {{a},{a,b}} just another way of writing {a,b}?