By Gerard O'Regan

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**Additional resources for Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications**

**Sample text**

A set S is a subset of a set T (denoted S T) if whenever s 2 S then s 2 T, and in this case the set T is said to be a superset of S (denoted T S). e. S = T if and only if S T and T S. A set S is a proper subset of a set T (denoted S & T) if S T and S 6¼ T. That is, every element of S is an element of T and there is at least one element in T that is not an element of S. In this case, T is a proper superset of S (denoted T ' S). 28 2 Sets, Relations and Functions T S The empty set (denoted by ∅ or {}) represents the set that has no elements.

He discovered the law of buoyancy known as Archimedes’s principle: The buoyancy force is equal to the weight of the displaced fluid. He is believed to have discovered the principle while sitting in his bath He was so overwhelmed with his discovery that he rushed out onto the streets of Syracuse shouting ‘Eureka’, but forgot to put on his clothes to announce the discovery. The weight of the displaced liquid will be proportional to the volume of the displaced liquid. Therefore, if two objects have the same mass, the one with greater volume (or smaller density) has greater buoyancy.

C. The Roman number system uses letters to represented numbers and a number consists of a sequence of letters. g. IX represents 9 and XL represents 40. g. MCC represents 1200. They had no zero in their number system (Fig. 10). The use of Roman numerals was cumbersome in calculation, and an abacus was often employed. An abacus is a device that is usually of wood and has a frame that holds rods with freely sliding beads mounted on them. It is used as a tool to assist calculation, and it is useful for keeping track of the sums and the carries of calculations.