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Gear300
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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 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}?
An ordered pair is a set of two objects or elements in a specific order. The first element is usually referred to as the x-coordinate and the second element as the y-coordinate. It is denoted as (x,y).
The main difference between an ordered pair and an unordered pair is the specific order in which the elements are listed. In an unordered pair, the order of the elements does not matter, while in an ordered pair, the order is important and changes the meaning of the pair. For example, the ordered pair (1,2) is different from the ordered pair (2,1).
An ordered pair is used in mathematics to represent the coordinates of a point on a coordinate plane. It is also used to represent the domain and range of a function, where the first element represents the input value (x-coordinate) and the second element represents the output value (y-coordinate).
No, an ordered pair cannot have duplicate elements. Each element in an ordered pair must be unique and cannot be repeated. This is because the order of the elements is important and changes the meaning of the pair.
A Cartesian product is the set of all possible ordered pairs that can be formed by combining elements from two sets. An ordered pair, on the other hand, is a specific set of two elements in a defined order. In other words, a Cartesian product is a collection of ordered pairs, while an ordered pair is a specific type of pair.