Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term "discrete mathematics" is therefore used in contrast with "continuous mathematics," which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus).
Discrete Mathematics in Education
Discrete mathematics is compulsory for most of those who are pursuing a college degree in computer science.Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in computer algorithms and programming languages, and have applications in cryptography, automated theorem proving, and software development. Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and logic, both of which are a part of discrete math.
Versus 1 - Real Number vs Distinct Values Continuous mathematics is, roughly speaking, mathematics based on the continuous number line, or the real numbers. The defining quality of it is that given any two numbers, you can always find another number between them. In fact, you can always find an infinite set of numbers between them. Building up from numbers, a function in continuous mathematics can take the form of a perfectly smooth curve without any gaps or breaks.
In discrete mathematics, you're working with distinct values - given any two points in discrete mathematics, there are not an infinite number of points between them. If you have a finite set of objects, you can describe the function as a list of ordered pairs, and present a complete list of those pairs.
The difference becomes clearer when you think about some of the deeper areas of math - which is sort of unusual. In general, getting deeper makes things harder; but here, getting deeper makes the difference easier to understand.
Versus 2 - Interval vs Natural Number
In continuous math, the fundamental set of numeric values that we use for proofs is the interval (0,1). We often prove various properties of sets by using mappings from values in the range (0,1).
In discrete math, the fundamental set of numeric values is the natural numbers, we prove properties of sets by using mappings from the natural numbers.
For example, one of the basic fundamental sets of concepts in mathematics consists of the ideas of shape, closeness, and adjacency.
Versus 3 - Topology vs Graph Theory
In continuous mathematics, we generally study those ideas using topology: sets of points form topological spaces, and we often study properties of those spaces and functions over them by using mappings from the range (0,1) to subspaces or functions.
In discrete mathematics, we study those ideas using graph theory, where we have a set of points where each point is connected to a specific set of other points by edges. We often study properties of graphs or functions over graphs using mappings from the natural numbers to subgraphs or functions.
Versus 4 - Thing Change vs Recurrence Relations
Another very fundamental thing we do in mathematics is study how things change. In continuous mathematics, we do that using differential equations, which are functions that describe the rate of change of one function using another derived function.
In discrete mathematics, we do the same thing using recurrence relations, which define a the value of a function at a point in terms of one or more of the points that precede it.
To Date - Discrete Mathematics
The study of how discrete objects combine with one another and the probabilities of various outcomes is known as Combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation. Topics in number theory such as congruences and recurrence relations are also considered part of discrete mathematics.
The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades.
To Understand the fundamental concepts of Discrete Mathematics vs Continuous Mathematics, let's try the following.
Differentiate between Them - Discrete or Continuous
For each of the following say whether the data is Discrete or continuous.
- Ages of people
- Goals scored in a school hockey match
- The number of votes in a council election
- The amount of water consumed by a household
- The viewing figures for a TV programme
- The time it takes to get home
- The number of stars that can be seen in the sky
- Ages of people - Continuous
- Goals scored in a school hockey match - Discrete
- The number of votes in a council election - Discrete
- The amount of water consumed by a household - Continuous
- The viewing figures for a TV programme - Discrete
- The time it takes to get home - Continuous
- The number of stars that can be seen in the sky - Discrete
