Discrete mathematics-first-order logic-Basic concepts

First order logic this piece belongs to the content of discrete mathematics, its function is to symbolize the natural things for the establishment of the system to lay the foundation of language.

Recall whether learning Chinese or English grammar, we are from the backbone of the sentence to learn, then mathematics as a language, its sentence of course there are so-called trunk. Individual words: Individual times are specific or abstract objects that the object of study can exist independently. Specific and specific object individuals become individuals often, usually in lowercase letters A, B, C is indicated. and the abstract or generalized individual words become individual variable, generally by the English letter x, Y, Z, and said that the individual variable value range is the individual domain.

To illustrate:

(1) "5 is the prime number", 5, the prime number are individual words, 5 is the individual constant term and the prime number is individual variable.

(2) "X>y", x, y are individual variable.

Predicate: Here seems to be similar to the natural language of the predicate verb, often described as "an action", but here, the predicate is described as "a relationship", of course, and the individual words are similar, according to the description of the relationship between individuals or not (specific or abstract generic), we can also divide the predicate into constant and variable.

To illustrate:

(1) x is the rational number. "is rational number" is a constant predicate.

(2) x has a specific relationship with y L. Here and its confusing is the statement "has a specific relationship L", but in essence the relationship L or abstract uncertainty, so here "has a specific relationship L" is a variable predicate. The next thing to do is to symbolize the statement that describes the relationship, which is somewhat analogous to the concept of a function, because the predicate describes the relationship between individuals, so it must depend on the individual. We use F, G, h to symbolize the representation.   F (a) and f (x) represent the properties F (a) and F (x) of the individual common item A, individual variable x, respectively. More generally, p (X1,X2,X3...XN) indicates that the individual x1,x2,... xn have a relationship P. For predicates that do not contain individual changes, we become a 0-dollar predicate.

EX1: Symbolize the following propositions in first-order logic with 0-tuple predicates and discuss their truth

(1) Only 2 is prime, and 4 is prime.

G (2) means that 2 is a prime number, G (4) means that 4 is a prime, then we symbolize the result of this proposition: g (2)-G (4), because the condition of the proposition is false, so the proposition is true.

(2) If 5 is greater than 4, then 4 is greater than 6

G (5,4) means "5 greater than 4", the result of the propositional symbolization: g (5,4), G (4,6), the condition is true conclusion is false, so the proposition is false.

Discrete mathematics-first-order logic-Basic concepts