Ordered Pair
The combination
of numbers or elements in a fixed order in such a way that the first component
always represents the x-component and second component always represents the
y-component is known as ordered pair. The elements of an ordered pair
are always separated by commas and enclosed within round or small brackets. Eg.
(1, 2), (p, q), (Sita, Ram) etc.
The
x-component is also known as antecedent and y-component is known as consequent.
Equality
of ordered pair
The ordered
pair (a, b) and (c, d) are said to be equal if their corresponding elements are
equal. i.e., the first component ‘a’ should be equal to the first component ‘c’
and the second component ‘b’ and ‘d’ should be equal.
Mathematically,
if (a, b) = (c, d), then, a = c and b = d.
Cartesian Product
Let A and B
be two non-empty sets. Then the product A × B (read as A cross B) is the set of
all ordered pairs in such a way that the first elements are always taken from
set A and second elements from set B.
Mathematically,
A × B = {(x,
y): x ϵ A and y ϵ B}
Let’s take
an example,
A = (1, 2)
and B = (3, 4)
Then A × B =
{(1, 3), (1, 4), (2, 3), (2, 4)}
This can be represented by an arrow diagram as follows.
A × B = {(1,
3), (1, 4), (2, 3), (2, 4)}
It can also be represented by a table.
A × B = {(1,
3), (1, 4), (2, 3), (2, 4)}
Similarly, B
× A can also be determined in the same way taking elements of set B first and
set A second.
Presenting B × A in a table.
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B × A = {(3,
1), (3, 2), (4, 1), (4, 2)}
Representing B × A in arrow diagram,
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