Professor Rotil entered the classroom and looked at the students. He was not particularly happy that morning, he never was happy in the morning. He looked at the students and asked, “quiet please”, the students collectively went into a quiet state immediately. Professor walked up to the blackboard, then, he kept down his books on the table in front of it. He looked at the class again. He asked, looking at the right hand side of the class in general, “where were we the last time”? No one answered.
A half minute went by.
Then, a hand came up, it was a female student from the second bench. “Yes”, came the word from the professor. The student got up from her seat and answered, “sets”. “Alright”, the professor acknowledged the answer, and motioned the student to sit down. The student sat down. Professor took half a minute and then turned towards the blackboard. He wrote, “Permutations” there.
“Alright, in the last lecture we must have seen the sets, which is a type of discrete mathematical structure. I repeat, the discrete mathematical structures are the ones which are not continuous. A range of real numbers between, say, 1 and 10 can be called continuous, as they can take on values like, 1.1, 1.12, 9.8, etc. There is no limit to their numbers. Discrete structures on the other hand, are not continuous, they can be counted and the count can be represented using a number”.
Professor Rotil looked at the class and continued. “A set having, say, 12 elements can be called a discrete structure. A set is a mathematical structure which has elements in an unordered fashion”. A hand rose up, from the first bench. It was a male student, named Kauti. “Yes”, the professor acknowledged the student. Kauti got up and asked, “are lists and arrays such structures too”? Professor sat him down with a motion of his hand and continued, “these are structures, but, these can be called computational structures rather than mathematical structures”. “The theory on this is loose and undefined, you are free to take a stab at it. It’ll be best, though, for the purpose of this class, to focus on the mathematics”. The professor then continued.
“Today, we are going to see permutations. This is another discrete mathematical structure. Again, this structure lacks the property of being continuous. Permutations can be defined as a set of elements taken from a set of elements, the order is important. Let’s say k elements taken from a set of n elements. Permutations count the number of such possibilities. There is a fixed formula for this”. Then, the professor started to write on the blackboard. The class, Kauti and the female student listened, some wrote things down in the notebook, the class was nearly quiet. The professor wrote the formula for permutations on the blackboard, some students copy it. The, he proceeded to explain the formula using an example. “Let’s say we have 10 people and 5 chairs. The problem is to find the number of ways in which the 5 people from this set of 10 people can be seated on the 5 chairs. We just have to apply the formula to calculate the number of ways”. Then, he applied the formula and wrote down the answer on the board. Some students copied it and some students just watched. The professor then looked at the class and let the students digest the lesson. He looked around, at the class, he took a minute.
“Next”, he started, “is the graph”, he said and started wiping the board, “the graph is another discrete structure”, he was done wiping the board and he started drawing on the board. He drew a graph on the board. “A graph is a set of nodes connected by straight lines. Here, this graph has 7 nodes and 8 connections. Again, this is a discrete mathematical structure owing to the fact that it has no continuous nature”. He continued, describing the graph.
At the back bench, a couple of students were completely lost, they had barely grasped the concepts of graph and permutations. They were lost in their own worlds. Their worlds did not cross into the world of the professor, other students or the discrete mathematical structures. They were 2 of them. One of them was sitting by the window. As the professor spoke, an image of his mother flashed in front of his eyes, and he got lost into what had happened that morning. He zoned out of the class and into his house. In a minute, the words of the professor which were related to mathematics became a lullaby for him and he fell asleep, by the window.
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