Chapter / 23ġ3 Translating English to Logic Translate the following sentence into predicate logic: Every student in this class has taken a course in Java. Richard Mayr (University of Edinburgh, UK) Discrete Mathematics.
DISCRETE MATHEMATICS ENSLEY CHAPTER 2 FREE
In particular, this expression contains a free variable. Chapter / 23ġ2 Precedence of Quantifiers Quantifiers and have higher precedence then all logical operators. The uniqueness quantifier can be expressed by standard operations.!x P(x) is equivalent to x (P(x) y (P(y) y = x)). Example: Let P(x) denote x 0 and U are the integers. The truth value depends not only on P, but also on the domain U. Chapter / 23ġ0 Existential Quantifier x P(x) is read as For some x, P(x) or There is an x such that, P(x), or For at least one x, P(x). If U is the positive integers then x P(x) is true. If U is the integers then x P(x) is false. Chapter / 23ĩ Universal Quantifier x P(x) is read as For all x, P(x) or For every x, P(x). (P(x) Q(y) P(z))) How many free variables does this formula contain? Clicker 1 One 2 Two 3 Three 4 Four 5 None Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 23Ĩ Quantifiers: Example Consider this formula Φ = y. A propositional function that does not contain any free variables is a proposition and has a truth value. All other variables in the expression are called free variables. Variables in the scope of some quantifier are called bound variables. The quantifiers are said to bind the variable x in these expressions. x P(x) asserts that P(x) is true for some x in the domain.
Symbol: x P(x) asserts that P(x) is true for every x in the domain. Symbol: Existential quantifier, There exists. The two most important quantifiers are: Universal quantifier, For all. Chapter / 23ħ Quantifiers We need quantifiers to formally express the meaning of the words all and some. Let Q(x, y, z) denote that x y = z and U be the integers. Chapter / 23Ħ Examples of Propositional Functions Let P(x, y, z) denote that x + y = z and U be the integers for all three variables.
Example: Let P(x) denote x > 5 and U be the integers. The domain is often denoted by U (the universe).
Chapter / 23ĥ Propositional Functions Propositional functions become propositions (and thus have truth values) when all their variables are either replaced by a value from their domain, or bound by a quantifier P(x) denotes the value of propositional function P at x. Variables stand for (and can be replaced by) elements from their domain. Can contain variables and predicates, e.g., P(x). Propositional functions are a generalization of propositions. Predicates (i.e., propositional functions): P(x), Q(x), R(y), M(x, y). Chapter / 23Ĥ Predicate Logic Extend propositional logic by the following new features. We need a language to talk about objects, their properties and their relations. Does it follow that Socrates is mortal? This cannot be expressed in propositional logic. Chapter / 23ģ Propositional Logic is not enough Suppose we have: All men are mortal. Chapter / 23Ģ Outline 1 Predicates 2 Quantifiers 3 Equivalences 4 Nested Quantifiers Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. 1 Discrete Mathematics, Chapter : Predicate Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics.
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