Would 2x10 joist be strong enough to support a living area of 13 foot span or should I go with 2x12. This will be a kitchen.

Based on the very limited info provided: 2 x 10 doug. fir #2 @ 16" o.c. joists should be fine for a "standard" kitchen at a 13' span. "Standard" kitchen would not have mud and tile floor and granite countertops and granite island. Also, this assumes no other bearing walls, etc. fall on the joists.

Thanks CT Engineer, there will be laminate wood floor and no granite countertops also no load bearing walls will fall on the joists. All cabinets, dishwasher, refridgerator will be on the outside walls so that the center of kitchen will be open.