Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20060423 16:32:15
Set TheorySet: A well defined collection of objects is called a set. If the element a is not present in set A, it is denoted by Description of a set I. Roster mehod: In this method, a list of all the objects of the set is made and they are put within braces. Example 1: If A is the set of the first eight prime numbers, A={2, 3, 5, 7, 11, 13, 17, 19} Example 2: If B is the set of all the counting numbers between 20 and 30, B={21, 22, 23, 24, 25, 26, 27, 28, 29} II. Set Builder form: In this method, the properties common to all the elemnts of the set are listed. It is written in the form{x:x has properties P} Example 1: If B={3,5,7,9,11}, it is written as B={x:x=2n+1 where n∈N, n<6} or B={xx=2n+1 where n∈N, n<6} Singleton set: A set consisting of a single elements is called a singleton set. Example : A={2} or A={x:x is an even prime number} Empty set: A set which has no elements is called an empty set. It is represented by {} or Finite and Infinite sets: A finite set is one in which the number of elements is finite. Example: A={x:x is a multiple of 5, x<1,000,000,000} An infinite set is one in which the number of elements is not finite. Examples: Set of all points on the arc of a circle, Set of all concentric circles with a given centre, {xx∈Q, 0<x<1} (Q is the set of Rational numbers). Equal sets: Two sets A and B are equal if every element of set A is in set B and everyelement of set B is in set A. It is denoted by A=B. Cardinal Number of a set:The number of (distinct)elements in a set is called the Cardinal number of the set. If A is the set, n(A) is the caridanl number of set A. Example: If A={0,1,2,3,4,5,6,7,8,9}, n(A)=10 Equivalent sets:Two sets are said to be equivalent if n(A)=n(B), that is, if their cardinal numbers are equal. Remark: Equal sets are always equivalent but equivalent sets are not always equal. Example: if A={1,2,3,4,5}, B={a,b,c,d,e} A and B are equivalent sets because n(A)=n(B)=5, but A and B are not equal sets. Subset: Let A and B be two sets given in such a way that every element of A is in B, then it is said that A is a subset of B, written as Superset:If A is a subset of B, then B is a superset of A, denoted by Proper subset: If A is a subset of B and set A is not equal to set B, then A is called a proper subset of B, denoted by Example, if A={1,2,3}, B={1,2,3,4,5,6,7}, then Comparable sets: Two sets A and B are comparable either if A is a subset of B or B is a subset of A. Properties of Subsets: 1. {}, the empty set, is a subset of all sets. 2. Every set is a subset of itself. 3. The number of all subsets of a set containing n elements is 4. The number of all proper subsets of a set containing n elements is 5. The set of all subsets of a given set A is called the Power set of set A, denoted by P(A). If A has n elements, P(A) has elements. Operation on sets Union of sets: The union of two sets A and B is the set of all eleements in A or in B or in both A and B. It is denoted by Example: if A={0,2,4,6,8,10} and B={1,2,3,4,5}, Intersection of sets: The intersection of two sets A and B is the set of elements common in A and B. It is denoted by If A={0,2,4,6,8,10} and B={1,2,3,4,5}, Difference of sets: The difference of two sets A and B is defined as the elements present in set A not present in set B. It is denoted by AB. If A={0,2,4,6,8,10} and B={1,2,3,4,5}, AB={0,6,8,10} and BA={1,3,5} It should be remembered that Symmetric Difference of sets: The symmetric difference of two sets A and B is defined as Symmetric difference is commutative. If A={0,2,4,6,8,10) and B={1,2,3,4,5}, AB={0,6,8,10}, BA={1,3,5} Universal Set: A set which is a superset of all the given sets, denoted by U, is known as the Superset. Complement of a Set: If set A is the subset of Universal set U, then the complement of A, denoted by is the set of all the elements in the Universal set not in A. Example: if U={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z} and A={a,e,i,o,u}, then Important results on Complements Laws of Operations Character is who you are when no one is looking. #2 20060423 17:24:45
Re: Set TheoryCommutative Laws: Associative Laws: Distributive Laws: De Morgan's Laws: Character is who you are when no one is looking. #3 20060423 21:54:17
Re: Set TheoryNumber of elements in a set Example Let As per the formula, Therefore, LHS=RHS. Character is who you are when no one is looking. #4 20060423 22:53:41
Re: Set TheorySome Important Results Character is who you are when no one is looking. #5 20060423 22:58:09
Re: Set TheoryCartesian Product Example: A={1,2,3}, B={a,b} Imortant Results of Cartesian products If A or B or both are infinite sets, A X B is also an infinite set. Character is who you are when no one is looking. #6 20061125 01:28:50
Re: Set Theoryhow can we say minimum no of elements of AUB IF N(A)=8 AND N(B)=5 #7 20070222 11:20:49
Re: Set Theoryyou said: " denotes an element is not in the set"this is wrong, the correct notation is Last edited by Sekky (20070222 11:28:13) #8 20070222 15:45:33
Re: Set TheoryHi Sekky, Character is who you are when no one is looking. #9 20070222 19:46:08
Re: Set Theory
No, it isn't, it's used to denote containment, not lack thereof. It's simply back to front, equally as every single other binary relation can be written back to front, as in subset containment or have you ever heard of < and >? The symbol means the same, but the relation is reflexed. #10 20070223 03:54:35
Re: Set TheoryActually this list should really be in the algebra formulas thread, since a set is just a degenerate algebra. #11 20070223 04:16:55
Re: Set TheorySekky, this is a big world with many, many, mathematicians. Is it really that much of a surprise that two different mathematicians don't use exactly the same symbols?
Can you please define "degenerate algebra"? Wikipedia, MathWorld, and myself have no idea what you mean when you say that. Also, if you're going to go with that notion, then every thread in the formula section should be put into Sets, as sets are pretty much the basis of all math. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #12 20070223 04:29:33
Re: Set Theory
He's using the same symbol to denote both the relation and the opposite relation, it makes no sense. The subset relation means the same back to front providing the relatives are switched, as does the less than symbol, as do any mathematical relations. #13 20070223 15:45:29
Re: Set TheorySekky, Character is who you are when no one is looking. #14 20070223 16:05:23
Re: Set Theory
A morphism is a mapping. A mapping is simply a special subset of AxB, where the mapping would be from A to B.
That's like saying: and Are the same symbol. I mean, they are just flipped around, right? "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #15 20070223 18:11:06
Re: Set Theory
Oh, right, we shifted into a parallel universe where everybody writes vertically. Last edited by Sekky (20070223 18:15:29) #16 20070223 23:24:00
Re: Set TheoryIn my experience there are alternative notations (which often lead to confusion, but that is how the world is). "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #17 20070224 02:29:56
Re: Set Theory
Whats a binary operator? Intersection and union or inclusion? Cause neither are. A binary operator is a mapping from AxA to A.
I will be doing so once this is resolved. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #18 20070224 11:13:11
Re: Set Theory
There's no way you can be THAT naive, you're obviously doing this on purpose, either for kicks or to chase me away from the forums, but I'll humour you in any case. #19 20070224 11:18:58
Re: Set Theory
Union an intersection each take two different sets. Not the same set, though they can be. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #20 20070224 11:23:28
Re: Set Theory
So you're saying any binary operator can only ever take the same element for it's operands? 2 + 3 = 5 last time I heard. #21 20070224 11:56:27
Re: Set Theory
No, but the elements must belong to the same set. In other words, the integers.
Yes, but this is restricting the definition of union because I can take: "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #22 20070224 12:06:49
Re: Set Theoryhence A and B are both elements of P{1,2,3,4,5,6} and no generality is lost, Z x Z > Z necessarily for some Z = P{1,2,3,4,5,6} Last edited by Sekky (20070224 12:10:43) #23 20070224 12:15:40
Re: Set Theoryor better, why not just consider the power set of the universal set under both operators, and I believe that forms a boolean lattice. #24 20070224 12:18:27
Re: Set Theory
Ah, I see. So now we're defining union by taking the union of two sets? And you don't see the problem in that? "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #25 20070224 12:22:09
Re: Set Theory
Union and Intersection are defined by PA x PA > PA 