2. Set Theory - Important Concepts

Key Takeaways

Hello dear students welcome to Dvikos commerce and management academy set theory introduction part is over now today we'll see important concepts of set theory or you can say it as a different types of sets total eight types of concepts we are going to learn important must focus very easy need not to worry so let's focus on each and every concept of this theory first one is that different types of test sets. First one finite and infinite sets. You know the meaning of finite. Finite means limited, countable, infinite, unlimited, uncountable. You know that the same meaning is applicable to the sets also. Now here first one finite you'll see if the number of elements are limited it is finite set. We know it. If the elements are limited, it is finite set. Example A is equal to elements are 1 2 3 4 5 easily we can find out five elements are there. It is a finite set. Okay. Where other side if the number of elements are unlimited? If it is limited. If it is unlimited infinite sit infinite sit for example fishes in a river can you count it? No stars in the sky infinite uncountable there's no end there. So this kind of elements whenever you find that is infinite set clear this is next coming to the null set. Null set we say it as empty set or void set also the symbol of null set is theta sin theta cos theta did you remember sameta you got it t theta means null set you can use this symbol theta or you can use this one also means empty brackets we are using but inside nothing is there it is also null set most of the time we use theta only you can see example Example, a set containing no element is called null set. A set containing no element. There is no element in the set. Then it is a null set. Example, you see A elements are given 1 2 3 4. B elements are 5 6 7 8 and they're asking us to find out A intersection B. If it is a union b means we would have written 1 2 3 4 5 6 7 8 all elements each and every element you could have taken. Now they are asking here A intersection B. A intersection means common numbers we have to take 1 2 3 4. Do you find 1 2 3 4 in the set of B? No. It means answer is nil. No, there is no answer. So that is why we can say it as a first we are taking the next step number two A intersection B we have written all the elements of A intersection and B elements. So nothing is matching. So the answer is TA or you can show empty bracket. This is null set. Null set. Okay. Next third one singleton set. Singleton set means a set containing a single element. A set is containing only single element. Example A. A set is having only one number that is six number. There is no other number. B set B it's containing only number 10. That's all. No other number means whenever you find only one number in one set that is singleton set. Easy to understand. Double T set means two two numbers triple T set three numbers can go a number like example singleton set. Next one equality of sets. Equality of sets means containing means when each element when each set elements are equal to other set. A element A set elements should be equivalent to the B set element. Then we can say equality of sets. This we can show with the symbol of A is equal to B. Equal A is equal to B or B is equal to A. Example A is here having sets A set is having elements of 1 2 3 4 whereas B is having 3 to 1 4. It's given in order but the B set is not in order doesn't matter but same numbers are there 1 2 3 4 1 2 3 4 elements are equal now then we can say A is equal to B that is equality of sets. Second example, A elements are given 0 1 3 4. B element is 0 3 2 5 but uh elements are not equal. Elements are not same. Here elements are may not be in proper order but elements are same. But now here elements are not same. So that is why we can say A is not equivalent to B. Here equality A is equal to B. Now A is not equal to B because elements are not matching. Next this is clear. Equality of sets. Then equivalent sets. Equivalent sets means when two sets are equivalent when two sets are equivalent. Set A is containing three numbers. Set B is also containing three numbers. Then we can say it as equivalent. Need not to be the same numbers. Just like here. Here the numbers must be same. Here the numbers need not to be same but numbers like three numbers are given. Here also three numbers are there. Then equivalent you can say example you can say if it is equalent then we can say a is equivalent to b. This is the symbol we use three is equals three lines we are using means this is equivalent or you can use another symbol also. Most of the time we use this symbol only. Example we are taking two examples. First example is A set is containing 1 2 three elements. B set is containing ABC. Introduction class I told you elements can be in alphabets or also can be numbers can be names also. So this is what we are taking out. A is 1 2 3 B is ABC. How many elements are there in A? Three. B how many elements here also? Three elements only. When elements are matching number of elements not exactly the number are alphabets number of elements 3 three it means then A is equivalent to B okay now second example I'm taking P is equal to 1 2 elements are given Q is given A DP means three elements are there here two elements and Q is given three elements it means P is not equivalent to Q. If it is equivalent, we are writing three lines. The same three lines if you go cross line means not equivalent. P is not equivalent to Q. Clear? Next one. Proper subset. Proper subset. Just focus here. Proper subset means if each element of A is an element of B, each element of A must be in B, then we can say proper subset. Proper subset we use this symbol C symbol C. Okay. Now example we are taking A set is having 1 2 3 4 B is having 1 2 3 4 5. What we said if each element of A is an element of B all the elements must be in the B set elements is it there 1 2 3 4 1 2 3 4 is here all these elements are available in B set yes this is a proper subset then A is proper subset of B. Okay. Now next another example P is equal to what are the elements are given P A B Q elements are B C D here only one two elements are there but these elements are are you finding in Q no finding it A is here A is not there [snorts] main A is missing so then A is not a proper subset of B if subset of B means C We are using proper not proper subset means cross same like not equivalent in the same way not a proper subset. Am I clear about it? Next power set important please focus power set it means the set of all possible subsets. The set of all possible subsets of set A is called power set of A. Set of all possible subsets. Please underline this point. The set of all possible subsets of set A. Set A. Set B is given. We are focusing on all possible subsets of subset A is called power set of A. This is we call it as a power set of A. symbol we use P power set of A or you can say 2 to the power of A. How many power sets are here? A means how many power sets are there? A is equal to number of sets. How many sets are there? Okay. Now let's focus on three examples. First example subset uh one one set a set is having only one number two. Then power set of a is equal to always you'll start with theta theta and then one only we we are having that I have written I have only one number only so I have written that so the elements which we are writing inside should be like elements only but outer side we are writing a big bracket so altogether I'm having only one element theta is nothing but nil only one element so I'm writing is 2 to the^ of 1 means one one shows number of elements how many elements I have only one so 2 to the^ of one 2 to the^ of 1 means two only I have written there second example a set is having two three elements means two elements are there elements are two 2 and three so total number is two 1 2 how do we Say power set of a power set of a is equal to we are writing first theta first we'll write two same like here 2 three also we are writing and combination of two and three another set so total how many sets we got 1 2 3 sets we got with two elements elements are two so we are writing 2 to the^ of two this two power of two represents How many elements are there? Earlier we had one element only. So we took 2 to the^ of 1 is equal to two. Now here 2 ^ of two because two elements are there. 2 ^ of two. 2 the^ of 2 means 2 into 2 4. Another example set of a is having 2 3 4. You want to find out power of element of a. So that is equal to theta. So you are writing first you are writing 2 3 4 then after that 2 3 2 4 then uh 2 4 then 3 4 we are writing and also there is a combination of 2 3 4 like so many are there. So this is 2 to the^ of three. Three items are there 2 to the^ of three. 2 ^ 3 means 2 into 2 into 2 2 into 2 4 4 into 2 8 so 8 this is power set and last one disjoint sets disjoint sets means if two sets A and B have no common element there's no common element no common element then A and B are disjoint disjoint sets. Say example A is having 1 2 3 set numbers. A is having elements of 1 2 3. B set is having 5 6 7. 1 2 3 5 6 7. There is no matching at all. It means disjoint. A and B are disjoint sets. Not matching each other. This is disjoint. Am I clear about it? Total we have learned today eight types of set. sets. Please note down clearly. Write down the notes. It will be useful. Short notes point of view, objective point of view, a multiplechoice point of view. You may get this kind of questions in any exam. Whether you may be having final exam or if you're writing any entrance test anywhere or competitive exams somewhere it will be useful. Write down it very clearly and understand and remember here. Don't forget to share this videos. Take out the screenshot. See you in the next video. Good luck.