The difference between simple proposition and compound proposition
Text/Li Sanping Luo Zengru
A section of "Simple Logic" has been added to the new textbook of Senior One. In the teaching process, both teachers and students have some difficulties and problems to varying degrees. For example, there are many different views on the distinction between "simple proposition" and "compound proposition". Even in the middle school mathematics education magazines, there are many debates on this issue, and it is difficult to form a unified understanding. We think this is mainly due to the lack of standards for distinction.
Understanding of the definition of 1
According to the definition of textbooks, propositions without logical conjunctions "or" and "not" are called simple propositions (those with logical books are called atomic propositions). It is considered that simple proposition is the most basic unit of logical calculus and should be regarded as an inseparable whole. For example, "3 is the divisor of 12" and "0.5 is an integer" are all simple.
A proposition composed of simple propositions and logical conjunctions is a compound proposition. For example, "20 is divisible by 4 or 5", "the opposite sides of a parallelogram are equal and parallel" and "2 is a non-prime number" are all compound propositions because they contain logical conjunctions or and respectively.
Several controversial examples.
From the definition of simple proposition and compound proposition to judgment and distinction, it seems easy to understand and master, but it is not. Please look at the following example.
Example: Explain whether the following propositions are simple or compound:
(1) "I will go to the classroom or library tomorrow morning";
(2) "A set of quadrilaterals with parallel and equal opposite sides is a parallelogram";
(3) "The square root of 4 is 2 or-2";
(4) "Two of the equations x2-5x+6 = 0 are x = 2 or x = 3";
(5) "The square of a real number is a positive number or 0".
These are several propositions that often appear in magazines and are controversial. Take proposition (3) as an example.
The first view is that proposition (3) is a simple proposition. This is because, if it is a compound proposition, there is
The square root of p: 4 is 2;
The square root of q: 4 is-2;
The square root of p or q: 4 is 2 or -2.
Because P and Q here are false propositions, it is incorrect to regard P or Q as an OR connection between P and Q according to the truth table. Therefore, proposition (3) is a simple proposition (here it is not related to whether P and Q are connected in the form of OR or not).
The second view holds that proposition (3) is a compound proposition. First, change the proposition (3) to its equivalent form, which can be written as: "The square root of 4 is 2 or the square root of 4 is -2". At this time, there is.
The square root of p:4 is 2;
Q: The square root of 4 is-2;
Square root of q or p: 4 is 2 or the square root of 4 is -2.
Because "P or Q" is the equivalent proposition of proposition (3), some articles think that proposition (3) is a compound proposition.
Similar to the second view, some authors think that proposition (3) is equivalent to "the square root of 4 may be 2 or the square root of 4 may be -2", so there is.
The square root of p:4 may be 2;
The square root of q:4 may be-2;
The square root of q or p: 4 may be 2 or -2.
Proposition (3) is a compound proposition, because it is equivalent to "P or Q" at this time.
So, is proposition (3) a simple proposition or a compound proposition?
3 criteria for distinguishing and judging
In our opinion, the controversy about the distinction of proposition (3) is mainly due to the lack of "standards" for distinction and judgment.
The discussion of a problem in mathematics can start from the form on the one hand and the essence on the other. For example, according to the definition of radical, we should say
As a radical, this is actually in form. After simplification, the result is 4 (essentially), which is an algebraic expression. Nevertheless, we still say it is a radical. For example, a large number of symbols have been introduced into mathematics:
Wait a minute. This is introduced for the convenience and conciseness of discussion.
But it is more important to understand and master its essence.
We believe that it is appropriate to take "essence" as the "standard" to judge and distinguish whether a proposition is a simple proposition or a compound proposition. One of the most important reasons is that it helps students to understand and master the proposition itself. According to this standard, we say that proposition (3) is a compound proposition. The second view is based on essence. What needs to be explained here is that proposition (3) is not the same as "the square root of 4 may be 2 or the square root of 4 may be -2". The word "possible" appears in the proposition, which makes it impossible to judge whether it is true or not within the scope of simple logic. A logic system that contains such logical constants as "necessity" and "possibility" is called "fuzzy logic". It can no longer be regarded as a proposition in simple logic. Similarly, the words "not necessarily" and "possible" in the proposition are beyond the discussion scope of simple logic.
The following are examples of several other propositions. Distinguish and judge according to the essence of "standard".
Analysis: Propositions (1) and (2) are simple propositions. Although they contain "or" and "and" respectively, they are not logical conjunctions, but should be regarded as conjunctions in natural language.
Logical conjunctions "or" and "qi" are similar in meaning to those in natural language, but they are not exactly the same. In simple logic, or is "either/or" (it is easy to know from the truth table), but in natural language, it often takes the meaning of "either/or". The "or" in the proposition (1) is "either/or". Therefore, the proposition (1) is a simple proposition. The meaning of "He" in Proposition (2) is the same as that of "He" in natural language, that is, only when "a group of opposite sides are parallel" and "Equality" exist at the same time, the quadrilateral is a parallelogram and inseparable, just like "He" in "Xiao Wang and Xiao Qiang are good friends".
The conditions and conclusions of proposition (4) are all open sentences, which are slightly different from those in simple logic, but they are still regarded as a proposition in simple logic without strict distinction here. Proposition (4) is essentially equivalent to "Equation X2-5x+6 = 0 has the root of x = 2 or Equation X2-5x+6 = 0 has the root of x = 3". Therefore, proposition (4) is equivalent to equation X2-5x+6 = 0.
Proposition (5) is also a compound proposition, which is completely expressed as "the squares of all real numbers are positive or 0". This proposition contains "quantifiers", which is different from the proposition discussed in simple logic. We will discuss such propositions in another article.
Taking the essence of a proposition as the "standard" for distinguishing and judging, one of the important steps is to first turn this proposition into its equivalent proposition. This criterion of distinguishing judgment is also applicable to some propositional forms that do not explicitly contain logical conjunctions.
For example, "3≥2", "24 is a multiple of 8 and 6" and "a triangle with two angles of 45 is an isosceles right triangle" does not contain logical conjunction, but it is equivalent to "3 > 2 or 3 = 2", "24 is a multiple of 8 and 6" and "has two angles" respectively.
refer to
1 Middle School Mathematics Room of People's Education Press. Full-time ordinary high school textbooks (trial revision? The first volume of mathematics (1) is a teacher's teaching book. Beijing People's Education Press 2000.
Gong Lei. On the "proposition" of learning and thinking. Middle school mathematics teaching reference, 2002, 9.
3 Xu Yanming. Analysis on the puzzle of proposition. Middle school mathematics teaching reference, 2002, 9.
4 Qin Qingyao, Zhang Dedong. Problems in the teaching of "Simple Logic". Middle school mathematics teaching reference, 2002, 9.