How to Calculate Proposition _ How to Calculate Proposition Assignment

What do propositions in discrete mathematics mean?

The following is the definition and basic explanation of the proposition. Understand the concept of proposition by yourself. To learn this chapter, we must first deeply understand the concept of proposition. Understand the relationship between atomic proposition and compound proposition, and understand the definition of conjunction on the basis of understanding compound proposition.

Proposition: a declarative sentence with a unique truth value is called a proposition, also called a statement for short. Note that there are two conditions here. First of all, it is a declarative sentence. Second, it has unique truth value.

Truth: The quality of being stated as true or false. The truth of a statement can be true or false. A true value does not mean that the value of a statement must be true.

Any proposition must have its truth value, which is the value of this proposition. Since it is a proposition, there must be a definite truth value, whether it is true or false. When a declarative sentence can tell whether its value is true or not (that is, it can always be one of them), it is a proposition, even if we don't know whether it is true or not.

In addition, we should understand the significance of propositional constant, propositional argument and assignment.

Compound proposition is a proposition that some atomic propositions are compounded by some conjunctions. Commonly used conjunctions are: (1) negation, (2) conjunction, (3) disjunction, (4) condition and (5) double condition.

Compound propositions are closely related to conjunctions. A proposition without conjunctions is an atomic proposition, and a proposition with at least one conjunctions is a compound proposition.

The truth value of compound propositions depends only on the truth value of each atomic proposition that constitutes them, and has nothing to do with their content and meaning. It has nothing to do with whether there is a relationship between two atomic propositions connected by conjunctions. This is very important, because when a proposition is expressed in natural language, we are often influenced by natural logic. For example, the proposition that it will rain if I don't go to work is untenable in natural logic. How can a person not go to work cause rain? But here, the value of this compound proposition is actually determined by the truth value of two atomic propositions, which has nothing to do with its meaning. This compound proposition is | p->; Q, the true value of the previous atomic proposition is false, and the value of the latter proposition is true. According to the definition of conditions, the value of this compound proposition is true. )

∧, ∨, ←→ are all symmetrical, and |→ are not symmetrical. (The textbook suggests that iff can also be used to represent two-way arrows ←→. Due to the limitation of character set, this page uses "|" when expressing negative related words. Please pay attention to the standard writing methods when writing. Symmetry refers to the relationship between the truth value of compound proposition and the truth value of atomic proposition in truth table. )

Propositional formulas are different from propositions. In a formula composed of proposition identifiers, if the identifiers represent definite propositions, the formula is a proposition. If the identifier only indicates the position of the proposition and can be replaced by any proposition, the formula is a propositional formula. When the propositional argument P is replaced by a concrete proposition, it is called the assignment of P.

Not all strings composed of propositional arguments, conjunctions and related brackets can become propositional formulas. To make a propositional formula (combination formula), it must conform to the regulations. This provision is:

(1) A single propositional argument is itself a compound formula.

(2) If A is compound, then |A is compound.

(3) If a and b are compound formulas, then (A∧B), (A∨B), (A → B) and (A→B) are compound formulas.

(4) If and only if (1)(2)(3) is limited, the symbol string containing propositional arguments, conjunctions and brackets is a compound formula.

It is generally understood that a single propositional argument is a compound formula, and as a propositional argument, a symbol string composed of conjunctions and brackets can only be a compound formula for a limited number of times. That is, the propositional formula, abbreviated as formula.

Propositional argument can only determine the true value of its propositional formula after assignment. When all propositional variables in a life formula specify a set of truth values, it is called assignment of propositional formula. Think about it, what is true assignment and what is false assignment? This is relatively simple.

The truth table of a proposition should list all its assignments. Generally speaking, the propositional formula consisting of n propositional arguments has 2n truth cases.

Simplification of conjunctions, according to two equivalent propositional formulas, we can see that a formula with more conjunctions can be simplified into a formula with fewer conjunctions. Here are two equivalent formulas that should be remembered:

(| P∨Q)& lt； = & gt(P→Q)

We should make clear what are tautology (eternal truth), contradiction (eternal fallacy) and satisfiability. This involves the value of assignment and propositional formula, which is easy to understand.