The most common form of mathematical proposition is p= >Q, or "if p is q", there are four forms of proposition, namely original proposition, inverse proposition, negative proposition and negative proposition. The relationship between them is as follows:
This is what we are familiar with. Now we use set theory to analyze the true and false relationship of the four propositions. Generally, let U={ all objects discussed}, let Ma={ objects with attribute a}, and let Mb={ objects with attribute b}. For example, when u = {quadrant}, Ma={ rectangle}, Mb = {parallelogram}, the four forms can be transformed into the inclusion relationship between corresponding sets: 1.
Original proposition, inverse proposition, no proposition, inverse proposition.
P = & gtq q = & gtp p = & gtq q = & gtp
MaMb MbMa cumacumb cumbCuma
Based on this, the relationship between propositions is transformed into the inclusion relationship between sets. And easy to understand:
1. When MaMb is established, there may be no MbMa, so when the original proposition is true, the inverse proposition may not be true, which is easy to understand. Similarly, the relationship between negative proposition and negative proposition is also the same.
2.MaMb is established, but cuMacuMb may not be established. Therefore, when the original proposition is true, the negative proposition is not necessarily true, which is also easy to understand. Similarly, the relationship between inverse proposition and negative proposition is also the same.
3. If MaMb is established, there must be cuMbCuMa, and vice versa, which shows that the original proposition is equivalent to its negative proposition.
4. if MaMb is established and MbMa is also established, there must be Ma=Mb, and of course CuMa=cuMb. In this way, when the original proposition and the inverse proposition are both true, the negative proposition and the inverse proposition are also true.
Based on the set representation of the relationship between the above propositions, it is easy to understand what positive results will be obtained after denying the conclusion of a proposition. For example, when proving by reduction to absurdity, if we use set tools to analyze the negation of conclusions, we can get better results.
What is the result of denying "A is a multiple of 3 or 5"? Students often say that "A is not a multiple of 3, nor is it a multiple of 5", which is wrong. We will analyze this proposition from the perspective of set. If a and b represent multiples of 3 and 5 respectively, then A∩B represents multiples of 3 and 5, that is, the set of numbers of multiples of 3 and 5. Obviously it is denied as cuA∩B, and according to Morgan's law, cuA∩B=cuA∪cuB.