Why is the air pressure of warm air lower than the air pressure of cool air?

The pressure exerted by an air mass is a function of its density; the greater the density, the greater the pressure. Warm air is less dense than cool air, so it exerts less pressure than cool air does. I hope that helps. Good luck!

Warm air is in an expanded state due to the heat. Cold air is contracted (denser). A cubic foot of warm air is not as dense as cold air and the cubic foot of cold air would weigh more than the warm air in that same cubic foot. Because cold air weighs more it is "pressing down" with more "pressure".

Its as simple as that!

See, the air pressure is inversely proportional to the temperature. So, when the pressure is high, the temperature is low and vice versa.

While comparing between warm air and cool air, the warm air has higher temperature than cool air. So, the pressure, being inversely proportional, is lesser in the warm air than that of it in the cool air.

Another good example for this (for better understanding of this concept) is that when water is cooled, i.e., the temperature is lowered, it becomes solid as a result of pressure(solid means its molecules have come closer coz there is increased pressure).

I like to think of it in terms of collisions between air molecules. Warmer molecules are moving faster so they hit the container's walls harder and exert more force on the wall.

But there is a complicating factor. Faster moving molecules will space themselves apart if they are allowed room to expand into, so there will be fewer of them impacting the wall in a given interval of time. That lowers the force exerted.

So things depend on whether you have your air trapped in a container or not.

when air is warm it expands and becomes lighter and excerts low pressure. when the air is cool it becomes heavy , densed and exerts high pressure ............. so, they r lighter than cool air

It ain't necessarily so. The density is less, because air expands when heated.

Cold air is heavier.