No, a plain can be a desert and a desert can be a plain. You can also have a desert that is not a plain and a plain that is not a desert. A plain is based on the structure of the land and a desert is based on the amount of precipitation it receives (low).
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In geography, a plain is land with relatively low relief, that is flat or gently rolling. Prairies and steppes are types of plains, and the archetype for a plain is often thought of as a grassland, but plains in their natural state may also be covered in shrublands, woodland and forest, or vegetation may be absent in the case of sandy or stony plains in hot deserts. Types of flatlands for which the term is not generally used include those covered entirely and permanently by swamps, marshes, playas, or ice sheets.
Plains occur as lowlands and at the bottoms of valleys but also on plateaus at high elevations. In a valley, a plain is enclosed on two sides but in other cases a plain may be delineated by a complete or partial ring of hills, by mountains or cliffs. Where a geological region contains more than one plain, they may be connected by a pass (sometime termed a gap). Plains may have been formed from flowing lava, deposited by water, ice or wind, or formed by erosion by these agents from hills and mountains.
Plains in many areas are important for agriculture, because where the soils were deposited as sediments they may be deep and fertile, and the flatness facilitates mechanization of crop production; or because they support grasslands which provide good grazing for livestock.
A desert is a landscape or region that receives an extremely low amount of precipitation, less than enough to support growth of most plants. Deserts are defined as areas with an average annual precipitation of less than 250 millimetres (10 in) per year,[1][2] or as areas where more water is lost by evapotranspiration than falls as precipitation.[3] In the Köppen climate classification system, deserts are classed as BWh (hot desert) or BWk (temperate desert). In the Thornthwaite climate classification system, deserts would be classified as arid megathermal