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Prove A union B minus A intersection B equals A minus B union B minus A?
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∙ 11y ago
complement of c
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Prove A union C minus B minus C equals A minus B union C?
This is really a version of deMorgans that states the complement of the intersection of any number of sets equals the union of their complements.We prove it in general and this is a specific case.Take x contained in the complement of the intersection of all sets Aj that is to say x is not in the intersection of all Aj . Now there must be at least one set that does not contain x since if all sets contain x then x would be in their intersection as well. Call this set A. Since x is not in A, x must be in the complement of A. But then x is also in the union of all complements of Aj , because A is one of those sets. This proves that the right hand set is contained in the left one.We now prove it the other way, that is to say, the left hand side is contained in the right. Remember that if A and B are sets and A is contained in B and B is contained in A we can say that as sets A=B.Consider x contained in the union of all complements of Aj . That means there is at least one complement that contains x, or in other words, at least one of the Aj does not contain x. But then x is not in the intersection of all Aj , and hence it must be in the complement of that intersection. That proves the other inequality, so both sets must be equal.
Is intersection of two topologies on X a topology on X?
A topology is a set of elements or subsets that follows these properties:∅ and X belongs to the setAny union of the subsets belongs to the set.Any intersection of the subsets belongs to the set.Yes, any intersection of two topologies on X is always a topology on X. Consider this example:Let X = {1,2,3}, T = {∅, {1},{2},{1,2},X} and S = {∅, {1}, {3}, {1,3},X}Then, S ∩ T = {∅, X, {1}}To show that S ∩ T is a topology, we need to prove these properties:∅ and X belongs to the setAny union of the subsets belongs to the set.Any intersection of the subsets belongs to the set.Step 1: Prove the first property is followedSince the empty set and X belongs to S ∩ T, the first property is followed. That is obvious. ;)Step 2: Prove the second property is followedSelect any union of any pair of subsets. You should see that this property is also satisfied. How?∅ U X = X ∈ S ∩ T∅ U {1} = {1} ∈ S ∩ T{1} U X = X ∈ S ∩ TStep 3: Prove the third property is followedAny intersection of the subsets belong to the set obviously. See below:∅ ∩ X = ∅ ∈ S ∩ T∅ ∩ {1} = ∅ ∈ S ∩ T{1} ∩ X = {1} ∈ S ∩ TSo the intersection of two topologies on X is a topology.
Is it true that if a union c equals b union c then a equals b?
No- this is not true in general. Counterexample: Let a = {1,2}, b = {1} and c ={2}. a union c = [1,2} and b union c = {1,2} but a does not equal b. The statement be made true by putting additional restrictions on the sets.
What Is partitioning in math?
Partitioning is dividing a set of things into subsets such that the union of all the subsets is the original set and the intersection of any two subsets is the null set. That is, between them, the subsets account for the whole of the original set and there are no elements in more than one subset.
What is a mathematical process in which sets are combined?
The union of two sets.The union of two sets.The union of two sets.The union of two sets.
Prove A union C minus B minus C equals A minus B union C?
This is really a version of deMorgans that states the complement of the intersection of any number of sets equals the union of their complements.We prove it in general and this is a specific case.Take x contained in the complement of the intersection of all sets Aj that is to say x is not in the intersection of all Aj . Now there must be at least one set that does not contain x since if all sets contain x then x would be in their intersection as well. Call this set A. Since x is not in A, x must be in the complement of A. But then x is also in the union of all complements of Aj , because A is one of those sets. This proves that the right hand set is contained in the left one.We now prove it the other way, that is to say, the left hand side is contained in the right. Remember that if A and B are sets and A is contained in B and B is contained in A we can say that as sets A=B.Consider x contained in the union of all complements of Aj . That means there is at least one complement that contains x, or in other words, at least one of the Aj does not contain x. But then x is not in the intersection of all Aj , and hence it must be in the complement of that intersection. That proves the other inequality, so both sets must be equal.
Prove if a union c equals b union c and a intersect c equals b intersect c then a equals b?
suppose x is in B. there are two cases you have to consider. 1. x is in A. 2. x is not in A Case 1: x is in A. x is also in B. then x is in A intersection B. Since A intersection B = A intersection C, then this means x is in A intersection C. this implies that x is in C. Case 2: x is not in A. then x is in B. We know that x is in A union B. Since A union B = A union C, this means that x is in A or x is in C. since x is not in A, it follows that x is in C. We have shown that B is a subset of C. To show that C is subset of B, we do the same as above.
What is the union intersection set?
Given two or more sets there is a set which is their union and a set which is there intersection. But, there is no such thing as a "union intersection set", as required for the answer to the question.
What is a antonym for intersection?
Union
Is there any case when union and intersection are same in sets?
if we have set A and B consider A={1,2,3,4}and B={3,4,5,6} the union of these sets is A and B={1,2,3,4,5,6}and the intersection is{3,4} the union and the intersection are same only if A=B
Do the intersection of two sets always contains a smaller number of terms than the union of two sets?
No, because the intersection of two equivalent sets will have a union the same size as its intersection.
If x and y and a be three non-empty sets such that a intersection x equals a intersection y and a union x equals a union y then prove that x equals y?
Let x, y, and a be sets and X,Y,x',y' be elements. Denote X *x as X in (is an element of) x, I as intersection, and U as union. If we can show that for all X *x, X *y (and similarly, if for all Y *y, Y *x), then we are done. Case 1) xIa is empty Then x, a and y, a have no elements in common. So, if xUa and yUa are equal, then for all y' *yUa but y' not*a, y' *y. Since xUa and yUa are equal, either y' *a or y' *x. But we supposed y' is not*a, so y'*x. Similarly, for all x' *x, x'*y. QED Case 2) xIa is non-empty Define a' as a - {x| x *xIa}. Then xIa' is empty, and you can use the same prove as above, replacing a with a'. QED
Difference between Union and Set Intersection Operation?
union means to group the given sets. where as intersection means to pick out the common elements from the given sets. if set a has 1,2,3 elements and B has 1,2,3,4,5. then its union will have 1,2,3,4,5 as its elements. and its intersection will have 1,2,3 as its elements.
What are the different sets operations?
Union, Intersection and Complement.
What are the basic operations for sets in math?
The basic operations are union and intersection.
What are set operations in math?
The basic operations are union, intersection and complement.
How do you use venn diagram to represent the union and intersection sets?
For two sets, the Venn diagram will consist of two overlapping ovals. The area of the overlap is the intersection. The entire area of both ovals is the union.
