Operations in the Set Theory

One of the theories used mathematics is the set theory. There are many operations available in it. There can be intersection and union. These are just two of the operations but there are many more. Union just gives a set containing all the elements present in the components.  The intersection of set gives a new set which contains the common elements of the components. There can intersection of three sets as well. In this the common elements of all the three are included in the final answer. Once this is found out the task becomes simpler as all common elements are known.

A convex set can also be defined in this concept. This is very closely related to set theory. In fact it is a part of the set theory. It is better to understand this concept. The intersection operation can take place on these also. The intersection of complex sets denotes all the points which will be available on the line which will be joining two points in vector space. The intersection of sets definition math can be explained with examples. The examples can make the concepts clearer. The examples are always useful in explaining the concepts in mathematics. The intersection operation basically helps in finding the common elements. So, this operation can be very helpful in finding the common elements. The other operations can also be very helpful in the set theory. The set theory has got a concept called Venn diagrams. They can be very helpful and explanatory. They can be used to solve the problems very easily. They are diagrammatically representation of the operations in the set theory. It is always better to use the diagrams as they can make the concepts clear and easily understandable.Having problem with how to round decimals keep reading my upcoming posts, i will try to help you.

So, to solve the problems in the set theory it is better to use the Venn diagrams. Venn diagrams can make the complex problems in mathematics into simpler ones. Once the problems are made simple, they can be solved. There can be various intersections like arbitrary intersection and nullary intersection. Null set is a set which does not contain any element. Universal set is the set which is the master set. There is one more important concept in the set theory. This is the concept regarding the disjoint set. In intersection there are common elements. But in the case of disjoint set there are no common elements. So, the answer is a null set.

Set Theory Venn Diagrams

INTRODUCTION TO SET THEORY VENN DIAGRAMS:

Define Set:

A set is a collection of well defined objects and these things which constitute a set are called its ‘elements’ or ‘members’.

Example:Set of odd numbers from 1 to 10 = {1, 3, 5, 7, 9}

Define Set theory - Venn diagram:

The geometrical representation of different types of sets is called “Venn diagram”

Example:

Basic Operations on Sets:

The primary operation on a set is,

- Union of sets:

The union of two nonempty sets A and B is the set consisting of all elements belonging to A or B or to both of them.

Ex: A = {1, 2, 5, 6} and B= {3, 7, 8, 9}

A `uu` B= {1, 2, 3, 5, 6, 7, 8, 9}

- Intersection of sets:

Let P and Q are two sets. The set of elements present in both the sets A and B are known as the set intersection of P and Q.

Ex P = {1, 2, 4, 6, 8}, Q = {6, 8, 9, 10, 12}.

P  `nn` Q = {6, 8}.

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Set Theorey Notations Used in Venn Diagrams

Let A, B are two sets and the following are the notations that are used in Venn diagrams:

AUB – >A union B which means that everything that is in either of the sets

A n B -> A intersect B which means that only the things that are in both of the sets

A c    -> A complement which means that everything in the object that is outside of A

A- B -> A minus B or A complement B which means everything in A except for anything in its overlap with B

- (AUB) -> not (A union B) which means everything outside of A and B

- (AnB) -> not (A intersect B) which means everything outer of the overlap of A and B



Set Theory Venn Diagrams Example Problem:

Mary asked 100 coffee drinkers whether they like cream or sugar in their coffee. 16 drinkers using cream only and 20 using both and 35 drinkers like sugar only.   How many like cream, sugar, cream but not sugar, sugar but not cream, cream and sugar, cream or sugar? Apply set theory Venn diagrams to solve.

Solution:We can find the solution for these questions using set theory Venn diagrams like

We should add 20 + 16 to find the no. of drinkers like cream

And 16+35 gives the number of sugar liking drinkers

Sugar but not cream is 35 and cream but not sugar is 16. Both gives 20 and we should add all 16+35+20 to find how many drinkers like cream or sugar which is 71.

Operations and Laws of Set Theory Venn Diagrams:

Operations on Sets:

Union of Sets
Intersection of sets
Disjoint of sets
Difference of two sets
Symmetrical difference of two sets
Complement of  a set
Algebraic laws:

Idempotent Laws
Identity laws
Associative laws
De-Morgan's Laws
Distributive laws