# A Practical Introduction to Data Structures and Algorithm by Clifford A. Shaffer By Clifford A. Shaffer

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It is poor programming practice to rely on the correspondence between 0 and false, because these are logically distinct objects of different types. Floor and ceiling: The floor of x (written x ) takes real value x and returns the greatest integer ≤ x. 0 = −3. The ceiling of x (written x ) takes real value x and returns the least integer ≥ x. 0 = −3. Modulus operator: The modulus (or mod) function returns the remainder of an integer division. Sometimes written n mod m in mathematical expressions, the syntax for the Java modulus operator is n % m.

Therefore, the result of n mod m must be between 0 and m − 1 when n and m are Sec. 3 Logarithms 31 positive integers. For example, 5 mod 3 = 2; 25 mod 3 = 1, 5 mod 7 = 5, and 5 mod 5 = 0. Unfortunately, there is more than one way to assign values to q and r, depending on how integer division is interpreted. The most common mathematical definition computes the mod function as n mod m = n − m n/m . In this case, −3 mod 5 = 2. However, Java and C++ compilers typically use the underlying processor’s machine instruction for computing integer arithmetic.

Note that P ∪ Q = Q ∪ P and that P ∩ Q = Q ∩ P, but in general P − Q = Q − P. In this example, Q − P = {10}. Note that the set {4, 3, 5} is indistinguishable from set P, because sets have no concept of order. Likewise, set {4, 3, 4, 5} is also indistinguishable from P, because sets have no concept of duplicate elements. The powerset of a set S is the set of all possible subsets for S. Consider the set S = {a, b, c}. The powerset of S is {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}. Sometimes we wish to define a collection of elements with no order (like a set), but with duplicate-valued elements.