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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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Discrete math is essential to college-level mathematics and beyond.

Discrete math is the mathematics of computing.

Discrete math is very much "real world" mathematics.

Discrete math shows up on most middle and high school math contests.

Discrete math teaches mathematical reasoning and proof techniques.

Discrete math is fun!

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GeoGebra-Integrated Professional Development of Future Math Teachers within Discrete Mathematics Course Valentyna Pikalova, G.S. Scovoroda Kharkiv National Pedagogical University, Ukraine Discrete Mathematics and adjacent disciplines are studied in all universities, where specialists in the field of programming, mathematics, as well as economic, technical and humanitarian areas are trained. Traditionally, discrete mathematics includes following topics such as set theory, mathematical logic, combinatorics, graph theory, etc. GeoGebra provides a wide range of instruments, which allow teacher to expand the range of tasks, increase their complexity, foster research component, create dynamic demonstration to explore mathematical objects with multiple representation. The complex of interactive dynamic demonstrations designed in GeoGebra to support discrete mathematics course for future teachers (pre-service) of mathematics is considered. Complex focuses on four types of demonstrations: teaching and learning basic concepts, stages of theorem proving, implementation of well-known algorithms from graph theory, the real-life problem-solving. This complex was created within the framework of future math teachers training. The course assignment involves student project on creating dynamic demonstration. ABSTRACT 2The computer simulation of SDN routing algorithm Liliia Bodnar, South Ukrainian National Pedagogical University named after K D Ushinsky Kateryna Shulakova, Odessa National Academy of Telecommunications named after A.S. Popov In this article, the authors review how the management and transmission of SDN elements can open new horizons for innovation. SDN is an architectural concept and a marketing term that combines a plurality of closed solutions building and network management. Existing simulation methods were reviewed for such networks. The mathematical software of environment GeoGebra can be offered as a simulation system of SDN routing algorithm. In a demonstration of Dijkstra's algorithm for different network models can clearly see the benefits that can be achieved using this algorithm in SDN networks. This research is aimed at simplifying network configuration, and as a result of reducing the cost of network equipment when not changing the technical specifications of nodes.

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GeoGebra-Integrated Professional Development of Future Math Teachers within Discrete Mathematics Course Valentyna Pikalova, G.S. Scovoroda Kharkiv National Pedagogical University, Ukraine Discrete Mathematics and adjacent disciplines are studied in all universities, where specialists in the field of programming, mathematics, as well as economic, technical and humanitarian areas are trained.

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Illustrating set operations with Venn Diagrams

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Illustrating set operations with Venn Diagrams

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Suitable for IGCSE set theory and A-Level S1. Clunky conditionals mean that selecting 3 sets will only load after a couple of seconds - apologies.

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JavaScript - Power Set - All Subset of S

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JavaScript - Power Set - All Subset of S

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From the diagram above, we can say that $a, b, c, d, e \in A$ since the elements $a$, $b$, $c$, $d$, and $e$ are contained within the set $A$. Furthermore, we can say that $f, g \not \in A$ since these elements are not contained within the set.

Now there are two ways to define a set. The first way is by explicitly writing our all of the elements in a set. For example, consider a set $A$ that lists all of the natural numbers less or equal to 5. Therefore, $A = \{ 1, 2, 3, 4, 5 \}$. Another way to define a set is by defining the set with a property $P(x)$, that is, $A = \{ x : P(x) \}$, which reads, "$A$ is the set containing elements $x$ such that $P(x)$ is true." In our example, we could write $A = \{ x : x \in \mathbf{N} \: \mathrm{and} \: x ≤ 5 \}$.

We also have some common symbols that denote special types of numbers:

Subsets and Supersets of a Set

Definition: A set $B$ is said to be a subset of $A$ written $B \subseteq A$ if all elements of $B$ are members of $A$, that is, if $b \in B$ then $b \in A$. Alternatively, we can say that $A$ is a superset of $B$ written $A \supseteq B$ to mean the same thing.

For example, suppose that $A = \{ 1, 3, 5, 7, 9 \}$ and $B = \{ 1, 3, 5 \}$. Since all elements of $B$ are also in $A$, we can say that $B \subseteq A$. In our diagram above, we note that $A = \{ a, b, c, d, e \}$ and $B = \{ b, d \}$ so $B \subseteq A$.

We should also note two other important definitions regarding sets very similar to the last definitions.

Definition: A set $B$ is said to be a proper subset of $A$ written $B \subset A$ if all elements of $B$ are members of $A$ but $B \neq A$. Similarly, we can say that $A$ is a proper superset of $B$ written $A \supset B$ to mean the same thing.

For example, consider the sets $A = \{ 1, 2, 3 \}$, $B = \{ 1, 2, 3 \}$ and $C = \{ 1, 2 \}$. We can say that $[[$ B$ and $C$ are subsets of $A$, that is $B \subseteq A$ and $C \subseteq A$. However, we cannot say that $B$ is a proper subset of $A$ since $A = B$. We can say that $C$ is a proper subset of $A$ though, that is $C \subset A$.

Equality of Sets

For example, consider the set of natural numbers $\mathbb{N} = \{ 1, 2, ... \}$ and the set of positive integers $\mathbb{Z}^+ = \{ 1, 2, ... \}$. We note that $\mathbb{N} \subseteq \mathbb{Z}^+$ and $\mathbb{Z}^+ \subseteq \mathbb{N}$. In other words, the set of natural numbers is fully contained in the set of positive integers and the set of positive integers is fully contained in the set of natural numbers. Therefore we can write that these sets are equal, that is $\mathbb{N} = \mathbb{Z}^+$. (Note that some people define the set of natural numbers to contain $0$ as well in which case these notations mean different things).

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From the diagram above, we can say that $a, b, c, d, e \in A$ since the elements $a$, $b$, $c$, $d$, and $e$ are contained within the set $A$. Furthermore, we can say that $f, g \not \in A$ since these elements are not contained within the set.