Regarding the immune system there are two types of lymphocytes-T-cells and B- cells, why are these so-named, and what is their respective function?

Hint: In the human body where the infection is defeated by a complex network of cells and proteins is called an immune system. Every microbe which enters the body will be noted by the immune system and this record will be useful in the future so that the immune system can defeat the germ whenever the same microbe enters the body again.

Complete answer:
Both T lymphocytes and B lymphocytes are generally white blood cells in a major way. They mostly are agranulocytes, definitely contrary to popular belief. Agranulocytes for the most part mean granulocytes which generally have no granules in their cytoplasm in a major way. They essentially are the sort of specific immune responders in a very major way. The very immune responders can kind of identify the invaders and attack, demonstrating how they are agranulocytes, which is quite significant. Both are produced by the bone marrow but mature in different places, showing how they kind of are agranulocytes in a big way. T lymphocytes can kill the infected cell, B lymphocytes produce antibodies that attack the antigens, which is fairly significant. T lymphocytes mature in the thymus; In nodes, B Lymphocytes are present, fairly further showing how the immune responders can identify the invaders and attack, demonstrating how they are agranulocytes, which is fairly significant.

Note:
In the immune system T-cells and B-cells have different responsibilities. Cell-mediated immunity and humoral immunity are the responsibilities for the T-cells and B-cells respectively. The common responsibility of both T-cells and B-cells is to identify the "non-self-antigens". This reorganization will be done during the antigen presentation.