
If d(x,y) is euclidean distance between x and y Prove that
d(x,y)>=0
if d(x,y)=0 than x=y
and d(x,y)=d(y,x)
Hi Velma,
The formula for d(x,y) has the form of the square root of a sum of squares.
For example, in 2dimensional euclidean space with x=(x_{1},x_{2}) and y=(y_{1},y_{2}),
d(x,y)=SQR[ (x_{1}y_{1})^{2} + (x_{2}y_{2})^{2} ]
 The square of a number is greater than or equal to 0, the sum of nonnegative numbers is nonnegative, and the square root of a nonnegative number is nonnegative so d(x,y) >= 0
 If d(x,y) = 0 then we have a sum of squares equal to 0 (*)
But squares are always nonnegative so (*) means each square must be zero.
Each square in the formula for d(x,y) represent the distance between x and
y when in each of the coordinate directions.
Since the distance between x and y in all coordinate directions is 0 it
must be that x and y are at the same location in euclidean space.
Therefore, x = y
 The formulas for d(x,y) and d(y,x) are only different in the subtraction
of the coordinates of x and y, where a negative sign appears (i.e., they
are the same in absolute value). But squaring eliminates the negative so
the sum of squares are the same.
Therefore, d(x,y)=d(y,x)
Cheers,
Paul
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