Restriction of a Set

• Apr 12th 2009, 03:44 PM
Snooks02
Restriction of a Set
My professor defined the restriction of a set:
R is a relation X. and Y is a subset of X.
all ordered pairs (x,x') that are elements of Y X Y such that xRx'.

What would be the restriction to the subset of positive real numbers for the relation xRy if x^2 + y^2 = 1?

I am kind of unsure exactly what this restriction means.
• Apr 13th 2009, 12:11 AM
Restriction of a set
Hello Snooks02
Quote:

Originally Posted by Snooks02
My professor defined the restriction of a set:
R is a relation X. and Y is a subset of X.
all ordered pairs (x,x') that are elements of Y X Y such that xRx'.

What would be the restriction to the subset of positive real numbers for the relation xRy if x^2 + y^2 = 1?

I am kind of unsure exactly what this restriction means.

Suppose X is the set {1, 2, 3, 4} and R is the relation on X: xRy if and only if x > y.

Then R = {(2, 1), (3, 1), (4, 1), (3, 2), (4, 2), (4, 3)}

Suppose now that we define a subset Y as {1, 2, 3}.

Then Y x Y = {(1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (3, 2), (1, 3), (2, 3), (3, 3)}

So the restriction of R to Y is those elements (ordered pairs) of R that are also elements of Y x Y; in other words {(2, 1), (3, 1), (3, 2)}.

So, in the question you are given, instead of all values of x and y (positive and negative) that satisfy $x^2 + y^2 = 1$ you need only those that are positive. On a Cartesian (x-y) plane, instead of the whole circle centre O, radius 1, we just get the quarter of this circle that lies in the first quadrant. OK?