When temperature goes up what happens to relative humidity?

For all other things unchanged:

Relative Humidity can be calculated by this equation: RH = (Vapor Pressure)/(Saturated Vapor Pressure) x100

From what we know about Saturated vapor pressures (saturated warm air has more water vapor than cool air that is saturated) we notice that more water vapor in the same given volume of air would inherently increase SATURATED vapor pressure. Thus, by simple mathematics (explained below), we can see that relative humidity would decrease as temperatures increase.

If, for a given fraction y/x, as x increases the fraction becomes smaller and smaller - assuming y is constant.