De Morgan’s laws, or theorems, are two principles of logic and set theory. What they establish and the demonstration of the first and second law of De Morgan.
Mathematics , as we know, is the subject most feared by students. The difficulty of the discipline, combined with the poor application of reasoning, represent two aspects that distance students from this school subject. However, the charm of mathematics is undeniable, especially if it is explained and understood clearly. Furthermore, a little effort and a lot of practice can make the difference.
This guide aims to illustrate De Morgan’s law , which takes its name from the mathematician Augustus De Morgan , who lived in the nineteenth century. This law finds application in the analysis of logic circuits .
In particular, De Morgan’s laws are two and are based on very simple formulas, which relate three set operations: union, intersection and complement .
De Morgan’s first theorem
De Morgan’s first law states that the complement of the intersection of two sets is equal to the union of the complements of the first and second sets .
Demonstration : To demonstrate this law, we need to draw two diagrams representing the sets A and B, together with their universal set. We represent the intersection of A and B, highlighting it with a color. Then, with different colors, we highlight respectively: the complement of the intersection of A and B, the complement of A and the complement of B.
We can observe that the union of the complements of A and B is the set of elements that belong to one of the two complements.
De Morgan’s second theorem
De Morgan’s second law states that the complement of the union of two sets is equal to the intersection of the complements of the first and second sets .
Proof : To demonstrate it, we need to draw the usual two diagrams representing the sets A and B, together with their universal set. Now we represent the union of A and B with a color. With different colors, we highlight respectively: the complement of the union, the complement of A and the complement of B.
We can note that the intersection of the complements of A and B is the set of elements that belong to both the complement of A and the complement of B.
The behavior of logical connectives
In practice, it can be said that De Morgan’s two theorems describe the behavior of logical connectives (known as AND and OR) when a negation is removed or inserted within a formula contained between parentheses.
If you collect the negation placed outside the brackets or distribute it inside the terms, the connective transforms into its opposite.
Finally, it should be remembered that these theorems can be proved both from an algebraic point of view and with the help of the truth table, since the cases to be proved can be expressed with a finite number.
Graphical representation of De Morgan’s law
It is recommended to use maximum precision in drawing the graphs to avoid confusion. It is very useful to use different colors to indicate the various steps, taking care to create a small legend that associates each color with the function it represents.
In textbooks you can find simple exercises to put these two laws into practice. The important thing is to memorize the steps described well and practice exposing all the main steps needed to explain De Morgan’s law.
