Let (X,d1) and (X,d2) be metric spaces. A.) Show that (X x Y, d) is a metric spa

Question

Let (X,d1) and (X,d2) be metric spaces.

A.) Show that (X x Y, d) is a metric space where d((x1,y1) , (x2,y2)) =the square root (d1(x1,x2))^2 + (d2(y1,y2))^2.

B.) Show that (X x Y, d) is a metric space where d((x1,y1) , (x2,y2)) = d1(x1,x2) + d2(y1,y2).

C.) Show that (X x Y, d) is a metric space where d((x1,y1) , (x2,y2)) =max{ d1(x1,x2) ,d2(y1,y2) }

PLEASE give details!! Many thanks.

Explanation / Answer

A)let z1=(x1,y1) and z2=(x2,y2) now,d(X x Y,d) is a metric space if 1)d(z1,z2)=0 if z1=z2 now,if z1=z2 then x1=x2 and y1=y2 and as,(X,d1) and (Y,d2) are metric spaces so d1(x1,x2)=0 and d2(y1,y2)=0 so, d(z1,z2)=sqrt(d1(x1,x2)^2+d2(y1,y2)^2)=sqrt(0+0)=0 So,1 holds 2) d(z1,z2)=d(z2,z1) as (X,d1) and (Y,d2) are metric spaces d1(x1,x2)=d1(x2,x1) and d2(y1,y2)=d2(y2,y1) so sqrt(d1(x1,x2)^2+d2(y1,y2)^2)=sqrt(d1{x2,x1)^2+d2(y2,y1)^2) so, d(z1,z2)=d(z2,z1) so,2 holds 3)d(z1,z2)+d(z2,z3)>=d(z1,z3) as (X,d1) and (Y,d2) are metric spaces d1(x1,x2)+d1(x2,x3)>=d1(x1,x3) and d2(y1,y2)+d2(y2,y3)>=d(y1,y3) therefore 3 follows. Hence it is a metric space Do others similarly.

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