My radiator cooling fan is running all of the time (even when car is off). I have been told it is likely the relay & know it is relay #13 or #15 per my manual & this post...

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Does the fan come on even with the car totally cold? If so I'd do your diagnostics with the car cold it will be easier.

Check Pin # 4 on the high speed fan relay & pin # 8 on the low speed. Those should be relay #13 & 15, respectively.

If its shorted to ground that means you are getting signal from the temperature switch, replace. If no short to ground then the relay itself is bad.
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How do I check the relays? I have a multimeter & I see the diagram on the side of the realy, but I don't know how to read it? Any help would be appreciated. Thanks.

A relay is a switch that is actuated by an electromagnetic coil. The diagram on the relay will show the coil and the switch. The switch on the diagram will be in the "normal" position, that is with the coil off.

Unplug the relay. If the fan is still on, there is a wiring problem. Check for continuity between the switch terminals on the relay. If you have continuity with the relay unplugged, the contacts are welded together and the relay should be replaced. If the relay clicks when you plug it in (cold engine) and the fan starts, then it is either wiring or the temperature switch.

Here is the relay that I replaced (high speed fan relay in position 13). It wasn't clear on how to test the switch based upon the diagram, but it fixed my problem. See below...

http://img143.imageshack.us/img143/8109/mg1570yy9.jpg

If someone would be so kind as to expalin the diagram & it would be greatly appreciated.