LESSON 1 - SETS and VENN DIAGRAMS



LESSON 1 - SETS and VENN DIAGRAMS

Main Lecture:-

Hey everyone! Are you ready for some fun-filled maths? Well since you have actually clicked on this video, it means that you really are ready! Anyway lets get started with our first topic. Its about Sets. This topic is a unique one, that is it cannot be regarded as your average maths topic involving unknowns are complex geometrical figures. Its more about logic and using your common sense.

1. Sets Definition and Notations

So then what is a set? A set is a well-defined collection of objects. Each object in a set is called an element of the set. The elements can be ANYTHING! (and of course the elements would give the Set its respective property.) For example if I talk about the Set of all positive even numbers below 10, my set would consist the elements : 2,4,6,8

Similarly if I talk about the set of lets suppose the great mathematicians of the past! Such a set would consist of elements : Carl Gauss, Isaac Newton, Einstein, Blaise Pascal, Euclid, Pierre de Fermat, etc..Sorry, I am not silly to list them all here!

There are other things as well that you should keep in mind: Two sets are equal if they have exactly the same elements in them; A set that contains no elements is called a null set or an empty set; If every element in Set A is also in Set B, then Set A is a subset of Set B.

Apart from these details you should also know that a set is usually denoted by a capital letter, such as A, B, or C, and that its elements are denoted by small letters for e.g x, y ,z

A set may be described by listing all of its elements enclosed in braces. For example, if Set A consists of the numbers 2, 4, 6, and 8, we may say: A = {2, 4, 6, 8}.

I have already told you what a null set is but you should also note that a null set is denoted by {∅} this symbol.

There also another very important set known as the epsilon set. It is denoted by 'є' . This actually the absolute or the universal set, that it covers all those sets which are given. For e.g if I talk about Sets A and B. Then these sets will actually be the subsets of the epsilon set. Epsilon set holds great importance for it can be used to impose restrictions so that the elements in the sub-set can be kept within limits. For e.g if I say the epsilon covers numbers 1 to 10 only. Then I add that Set A (part of that epsilon) is made up of odd numbers. Its elements would be : 1,3,5,7,9 . It would stop at 9 even though 11,13 onwards are also odd numbers. Why? Because you have to stay within the limits of the epsilon!

Sets may also be described by stating a rule. We could describe Set B {2,4,6,8} by stating: Set B consists of all the even single-digit positive integers.

2. Set Operators

Suppose we have four sets - W, X, Y, and Z. Let these sets be defined as follows: W = {2}; X = {1, 2}; Y= {2, 3, 4}; and Z = {1, 2, 3, 4}.

• The union of two sets is the set of elements that belong to one or both of the two sets. Thus, set Z is the union of sets X and Y.

• Symbolically, the union of X and Y is denoted by X ∪ Y.

• The intersection of two sets is the set of elements that are common to both sets. Thus, set W is the intersection of sets X and Y.

• Symbolically, the intersection of X and Y is denoted by X ∩ Y.

3. Sets and Venn Diagrams (the inseparable!)

Now lets talk about Venn Diagrams. To define, A Venn diagram is a way of representing sets visually.

To explain, we will start with an example where we use whole numbers from 1 to 10.

We will define two sets taken from this group of numbers:

Set A = the odd numbers in the group = { 1 , 3 , 5 , 7 , 9 }

Set B = the numbers which are 6 or more in the group = { 6 , 7 , 8 , 9 , 10 }

Some numbers from our original group appear in both of these sets. Some only appear in one of the sets.

Some of the original numbers don't appear in either of the two sets. We can represent these facts using a Venn diagram.

|[pic] |The two large circles represent the two|

| |sets. The rectangle represents epsilon.|

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|The numbers which appear in both sets|[pic] |

|are 7 and 9. These will go in the | |

|central section, because this is part| |

|of both circles. | |

|The numbers 1, 3 and 5 still need to | |

|be put in Set A, but not in Set B, so| |

|these go in the left section of the | |

|diagram. | |

|Similarly, the numbers 6, 8 and 10 | |

|are in Set B, but not in Set A, so | |

|will go in the right section of the | |

|diagram. | |

|The numbers 2 and 4 are not in either| |

|set, so will go outside the two | |

|circles. | |

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|The final Venn diagram looks like | |

|this: | |

|We can see that all ten original | |

|numbers appear in the diagram. | |

|The numbers in the left circle are | |

|Set A | |

|{ 1 , 3 , 5 , 7 , 9 } | |

|The numbers in the right circle are | |

|Set B | |

|{ 6 , 7 , 8 , 9 , 10 } | |

The intersection of sets A and B is those elements which are in set A and set B. A diagram showing the intersection of A and B can be seen below:

[pic]

The union of sets A and B is those elements which are in set A or set B or both. A diagram showing the union of A and B can be seen below:

[pic]

QUESTION/ANSWER SESSION:-

Q1. Describe the set of vowels.

Ans. If A is the set of vowels, then A could be described as A = {a, e, i, o, u}.

Q2. Describe the set of positive integers.

Ans. Since it would be impossible to list all of the positive integers, we need to use a rule to describe this set. We might say A consists of all integers greater than zero.

Q3. Set A = {1, 2, 3} and Set B = {3, 2, 1}. Is Set A equal to Set B?

Ans. Yes. Two sets are equal if they have the same elements. The order in which the elements are listed does not matter.

Q4. What is the set of men with four arms?

Ans. Since all men have two arms at most, the set of men with four arms contains no elements. It is the null set (or empty set).

Q5. Set A = {1, 2, 3} and Set B = {1, 2, 4, 5, 6}. Is Set A a subset of Set B?

Ans. Set A would be a subset of Set B if every element from Set A were also in Set B. However, this is not the case. The number 3 is in Set A, but not in Set B. Therefore, Set A is not a subset of Set B.

Q7. [pic]

(a) Which numbers are in the union of A and B?

Ans. [pic]

(b) Which numbers are in the intersection of A and B?

Ans. [pic]

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