Negation in Discrete mathematics
To understand the negation, we will first understand the statement, which is described as follows:
The statement can be described as a sentence that is not an exclamation, order, or question. A statement will be acceptable only if it is either always false or always true. Sometimes we want to find out the opposite of the given mathematical statement. In this case, the negation will be used. So, the negation of a statement can be described as the opposite of a given statement.
Negation
In discrete mathematics, negation can be described as a process of determining the opposite of a given mathematical statement. For example: Suppose the given statement is “Christen does not like dogs”. Then, the negation of this statement will be the statement “Christen likes dogs”. If there is a statement X, then the negation of this statement will be ~X. The symbol “~” or “¬” is used to represent the negation. So if we have a statement that is true, then the negation of this statement will be false. In contrast to this, if we have a statement that is false, then the negation of this statement will be true.
In other words, negation can be described as a refusal or denial of something. If your sister thinks you are a liar and you say that you don’t, this statement will be a negation. There can also be other negation statements like “I don’t kill my wife” and “I don’t know the name of that girl”. When we try to find the opposite meaning of a particular statement, then we can easily do this by inserting a negation. The words of negations can be ‘not’, ‘no’ and ‘never’. For example, we can do the opposite of the statement “I am playing” just by saying “I am not playing”.
If we do negation of the negated statement, then the general statement will be the original statement. We will understand this concept by an example, which are described as follows:
- Here, we will assume a statement, “The population of India is very big”, which is represented by X.
- Thus, the negation of a given statement will be “The population of India is not very big”, which is represented by ~X.
- The negation of the above-negated sentence will be “The population of India is very big”, which is represented by ~(~X).
Hence, it is proved that the negation of negated statement will be the given original statement.
Rules to get the Negation of statement
There are various rules to get the negation of statement, which are described as follows:
First, we have to write the given statement with the word ‘not’. For example, the multiplication of 3 and 5 is 15. The negation of a given statement is “the multiplication of 3 and 5 is not 15”.
If we have those types of statements that contain “All” and “Some”, then we have to make suitable modifications. For example: “Some people are not religious”. The negation of this statement is “All people are religious”.
Negation of X or Y
For this, we will assume a statement, “We are either Bania or Healthy”. This statement will be false if we can’t be bania and we cannot be healthy. The opposite of this statement is to be not Bania and not Healthy. Or if we want to rewrite this statement in the form of original statement, then we will get “We are not Bania and not Healthy”.
If we assume the statement “We are Bania” as X, and another statement “We are Healthy” as Y, then the negation of X and Y will be the statement “Not X and Not Y”.
In general terms, we will also get the same statement, i.e., The negation of X and Y is the statement “Not X and Not Y”.
Negation of X and Y
Here we will also take an example to understand this. For this, we will assume a statement, “We are both Bania and Healthy”. This statement will be false if we could be either not Bania or not Healthy. If we assume a statement “We are Bania” as X, and another statement “We are Healthy” as Y, then the negation of X and Y will be the statement “We are not Bania or we are not Healthy”, or “Not X or Not Y”.
Negation of “If X, then Y”
We can use another statement, “X and Not Y” in place of the statement “If X, then Y” so that we can do negation of X and Y. In starting, this replaced statement seems confusing. To understand this, we will take a simple example, which will help us to know why this is the right thing to do.
For this, we will assume a statement, “If we are bania, then we are healthy”. This statement will be false if we need to be bania and not healthy. If we assume a statement “We are bania” as X, and another statement “We are Healthy” as Y, then the negation of X and Y (X ⇒ Y) will be the statements, “We are Bania” = X, and “We are not Healthy” = not Y. In conclusion, the negation of “If X, then Y” becomes “X and not Y”.
For example: In this example, we will consider a statement of mathematics. So we will assume a statement, “If n is even, then n/2 is an integer”. If we want to show this statement to be false, then we want to determine some even integer n for which n/2 was not an integer. So we can say that the statement “n is even and n/2 is not an integer” is the opposite of the given statement.
Negation of “For every …”, “There exists….”
In discrete mathematics, sometimes we use the phrases like “for every”, “for all”, “for any”, and “there exists”.
For this, we will assume a statement “For all integers n, either n is even or odd”. This phrase is a little bit different from the other one, which we have learned above. This statement can be described in the form “If X, then Y”. The above statement can be reworded like this “If n is any integer, then either n is even or odd”.
If we want to determine the opposite/false of this statement or negate this statement, then we have to determine an integer that will be not even and not odd. There are some other ways in which we can describe this statement like this “There exists an integer n, so that n is not even and n is not odd”.
If we are negating a statement that is involved with the phrases “for all”, “for every”, in this case, this phrase will be replaced with “there exists”. Similarly, when we are negation a statement that is involved with the phrase “there exists”, in this case, this phrase will be replaced with “for all”, “for every”.
Example:
In this example, we will consider a statement “If all the bania people are healthy, then all the Punjabi people are thin”. To understand this, we will assume a statement “If all the bania people are healthy” as X, and another statement “all the Punjabi people are thin” as Y. We will assume this statement in the form “If X, then Y”. So the negation of this statement will be in the form “X and not Y”. So we can say that we need to negate Y. So the negation of Y will be the statement, “There exists a Punjabi person who is not thin”.
When we put these statements together, we will get “All bania people are healthy, but there exists a Punjabi person who is not thin” as the negation of “If all bania people are healthy, then all Punjabi people are thin”.