By Vivaldi F.
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BAD: The set which contains one and only one integer. As a final exercise, we express some elementary geometric facts using set terminology. The intersection of a line and a conic section has at most two points. The set of rational points in an open interval is infinite. A cylinder is the cartesian product of a segment and a circle. There is no finite partition of a triangle into squares. 2. FUNCTIONS 23 The intersection of a nested set of open discs may be empty. We encourage the reader to re-visit known mathematics from an advanced standpoint, using set language.
This definition requires a value for gcd(0, 0), which normally is taken to be zero. Let f : A → B be a function. 14) is called the graph of f . So a function is completely specified by three sets: domain, co-domain and graph. We can now re-write the definition of a function, disposing of the rather vague term ‘rule’ appearing in our original definition, and replacing it with a graph. We write a formal definition, and use a layout appropriate for it. D EFINITION. A function f is a triple (X ,Y, G) of non-empty sets.
The operator NOT is unary, that is, it takes just one boolean operand and produces a CHAPTER 3. LOGICAL STRUCTURES 52 boolean result. In this respect the operator NOT resembles the unary arithmetical operator ‘−’, which changes the sign of a number. The operator AND is binary —it acts on two operands and produces a boolean result. Think of it as a kind of ‘multiplication’. 7) The expression ¬P is called the negation of P. 5). Thus ¬(x < y) = (x y), ¬(x ∈ A) = (x ∈ A), ¬(A ⊂ B) = A ⊂ B. If P and Q are boolean expression, then P ∧ Q is called a compound expression, or compound statement.
Mathematical writing. An undergraduate course by Vivaldi F.