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Set theory activity

Set theory activity


Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. How to use Sets in Math?

We often deal with groups or collection of objects in real life, such a set of books, a group of students, a list of states in a country, a collection of baseball cards, etc.

set theory activity

Sets may be thought of as a mathematical way to represent collections or groups of objects. The concept of sets is an essential foundation for various other topics in mathematics. This series of lessons cover the essential concepts of math set theory - the basic ways of describing sets, use of set notation, finite sets, infinite sets, empty sets, subsets, universal sets, complement of a set, basic set operations including intersection and union of sets, using Venn diagrams and simple applications of sets.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other.

We can use these sets understand relationships between groups, and to analyze survey data. An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set. A set is a collection of distinct objects, called elements of the set. A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets. A set simply specifies the contents; order is not important.

Commonly, we will use a variable to represent a set, to make it easier to refer to that set later. Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. A subset of a set A is another set that contains only elements from the set Abut may not contain all the elements of A. A proper subset is a subset that is not identical to the original set—it contains fewer elements.

C is not a subset of Asince C contains an element, 3, that is not contained in A. There are many possible answers here.

Set theory

One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature. Commonly sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends.

At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets. The union of two sets contains all the elements contained in either set or both sets. The intersection of two sets contains only the elements that are in both sets.

The complement of a set A contains everything that is not in the set A.Which of the following are sets?

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Write the following sets in Set-Builder form. Write the following sets in Roster form. Write the following sets in Descriptive form. Determine whether B is a proper subset of A.

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In the given sets A and B, every element of B is also an element of A. But B is equal A.

set theory activity

Hence, B is the subset of A, but not a proper subset. Let us look at the next problem on "Set theory worksheets". And also But B is not equal to A. Hence, B is a proper subset of A. Hence, the number of proper subsets of A is The formula for cardinality of power set of A is given below. Here "n" stands for the number of elements contained by the given set A. Hence, the cardinality of the power set of A is Find how many had taken one course only. Find the total number of students in the group.

Assume that each student in the group plays at least one game. Find the number of students who like. Let M and S represent the set of students who like math and science respectively.

From the information given in the question, we have. After having gone through the stuff given above, we hope that the students would have understood "Set theory worksheets".First we specify a common property among "things" we define this word later and then we gather up all the "things" that have this common property. There is a fairly simple notation for sets.

set theory activity

We simply list each element or "member" separated by a comma, and then put some curly brackets around the whole thing:.

The three dots OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic.

Who says we can't do so with numbers? There can also be sets of numbers that have no common property, they are just defined that way. For example:. Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are. Math can get amazingly complicated quite fast. But there is one thing that all of these share in common: Sets.

We call this the universal set. It's a set that contains everything.

Discrete Mathematics/Set theory/Exercises

Well, not exactly everything. Everything that is relevant to our question. In Number Theory the universal set is all the integersas Number Theory is simply the study of integers. But in Calculus also known as real analysisthe universal set is almost always the real numbers. Also, when we say an element a is in a set Awe use the symbol to show it. And if something is not in a set use. Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!

Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal! Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A. Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them. By pairing off members of the two sets, we can see that every member of A is also a member of B, but not every member of B is a member of A:.

If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.Set theory is a branch of mathematical logic that studies setswhich informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

The language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the s.

After the discovery of paradoxes in naive set theorysuch as Russell's paradoxnumerous axiom systems were proposed in the early twentieth century, of which the Zermelo—Fraenkel axiomswith or without the axiom of choiceare the best-known. Set theory is commonly employed as a foundational system for mathematicsparticularly in the form of Zermelo—Fraenkel set theory with the axiom of choice. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

Mathematical topics typically emerge and evolve through interactions among many researchers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century.

Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kroneckernow seen as a founder of mathematical constructivismdid not.

Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" " Cantor's paradise " resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came aroundwhen it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes.

Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox : consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself.

In Cantor had himself posed the question "What is the cardinal number of the set of all sets? Russell used his paradox as a theme in his review of continental mathematics in his The Principles of Mathematics. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment.

The work of Zermelo in and the work of Abraham Fraenkel and Thoralf Skolem in resulted in the set of axioms ZFCwhich became the most commonly used set of axioms for set theory.

The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics.

Set theory is commonly used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology— category theory is thought to be a preferred foundation.

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Set theory begins with a fundamental binary relation between an object o and a set A. Since sets are objects, the membership relation can relate sets as well.

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A derived binary relation between two sets is the subset relation, also called set inclusion. As insinuated from this definition, a set is a subset of itself.

For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined.

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Just as arithmetic features binary operations on numbersset theory features binary operations on sets. Some basic sets of central importance are the empty set the unique set containing no elements; occasionally called the null set though this name is ambiguousthe set of natural numbersand the set of real numbers. A set is pure if all of its members are sets, all members of its members are sets, and so on.

In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchybased on how deeply their members, members of members, etc.

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams.

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The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition.Attribution theory holds that people naturally want to assign a reason for their successes and failures.

The reasons they choose have a significant impact on their future performance. When a student fails a test, for example, she is more likely to do better on the next test if she thinks she didn't study enough rather than if she blames her teacher.

Classroom activities using attribution theory can show how expectations can become self-fulfilling prophecies. In a study published in the "Journal of Personality and Social Psychology," researchers used attribution theory in a fifth-grade classroom to change student behavior.

First, the researchers handed out candies wrapped in plastic to the class just before recess. After the students left, they counted the number of wrappers on the floor and in the trash can. For the next two weeks, the teacher, the principal and others praised the students for being neat.

The researchers visited the classroom a second time and passed out wrapped candies. This time, they discovered a lot more wrappers in the trash than on the floor. They concluded they had achieved this desired result simply by changing the students' expectations of themselves. The students believed they were neat, so they became neater.

In a separate study published in the same issue of the "Journal of Personality and Social Psychology," the same researchers tested attribution theory using before-and-after measurements of math achievement and self-esteem.

They developed scripts for the teachers to use with each student. The scripts provided attribution training, persuasion training or reinforcement training. The attribution script told students they were working hard at math and to keep trying. The persuasion training essentially told the students that they "should" be good at math. The reinforcement training used phrases such as "I'm proud of your work" and "excellent progress. The explanation, the researchers concluded, is that students who received attribution training attributed their math performance to their own hard work.

This motivated them to work harder, and their results improved. Attribution theory supports the view that only students who think they are good spellers are motivated by spelling bees. Knowing this, teachers can structure spelling bees to motivate students who are not likely to win the competition.

A team spelling competition, in which evenly matched the teams contain both strong and poor spellers, can motivate spellers of all abilities by making them believe they have a chance to win. Structuring spelling competitions so that students spell words that match their abilities provides a more attainable — and motivational — goal. Awarding students for reaching a high level of achievement, such as 90 percent of words spelled correctly, engages a greater number of students by providing the expectation that they can achieve success.See more testimonials Submit your own.

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Refine Your Results. Content Curators. Resource Types. What Members Say. Get Free Trial. We found 1, reviewed resources for set theory. Lesson Planet. For Teachers 9th - 12th Standards. Introducing the vocabulary and concepts from ground zero and building to more complex ideas of subsets and Get Free Access See Review.

For Teachers 9th - 12th. Students compare the evolution theories of Lamarck and Darwin. They use self-assessment and a video to increase their knowledge of evolution theories. They research questions and present them to the class.

For Teachers 6th - 12th Standards. The heartbreaking story of Alfred Wallace's loss of collected evidence opens this documentary about the development of the theory of evolution. You will find supportive resources to use with the movie in your biology class. For Students 9th - 12th. Take a close-up look at the evolution of hyenas in South Africa. Natural historians read about the five hyena species found in the fossil record and examine four statements that summarize the theory of evolution.

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As a culminating For Teachers 6th - 8th. Music theory lessons can be very tricky for some people. Children with a basic understanding of musical concepts take on the task of transposing music and identifying scale sets. This would be a good topic to address prior to discussing For Teachers 8th - 12th. Students analyze 5 separate theories of evolution in order to help them explain the different meanings of theory, how human values influence science, and that the scientific view of the origin of life does not involve supernatural forces.

For Teachers 10th - Higher Ed.