How Can You Prove Set Equality for (A-B) ∩ C and (A ∩ C) - B?

In summary, an intersection in mathematics is the set of elements shared by two or more sets, denoted by the symbol ∩. To prove that two sets have an intersection, at least one common element must be present in both sets. An example of a proof involving intersection is using the element 2 to show that the sets A = {1, 2, 3} and B = {2, 3, 4} have an intersection of {2}. In set theory, the union and intersection are complementary operations, with the union combining all elements and the intersection finding only common elements. In real life, intersection is used in various fields such as economics, statistics, and computer science for analyzing overlapping markets, comparing data sets
  • #1
chica911
1
0
1. Let A , B and C be sets. Prove that (A-B) ∩ C = (A ∩ C)- B = (A ∩ C) – (B ∩ C).


3. Proof:
1st part: Let A, B and C be sets where (A-B) ∩ C. Let X be a particular, but arbitrary element of C. Since C and (A-B) ∩, X € (A-B) and X € C. Therefore, X € A but X ∉ B. Since X is an element of A and C, A ∩ C and since X ∉ B, (A ∩ C)-B. Therefore (A-B) ∩ C=(A ∩ C)-B.
2nd part: Let A, B and C be sets where (A ∩ C)- B.
3rd part: Suppose (A-B) ∩ C=(A ∩ C)-B and (A ∩ C)- B = (A ∩ C) – (B ∩ C) By the transitive property (A-B) ∩ C=(A ∩ C) – (B ∩ C) and therefore (A-B) ∩ C = (A ∩ C)- B = (A ∩ C) – (B ∩ C).

The 2nd part is what I need help with. I am not sure how to prove (A ∩ C)- B = (A ∩ C) – (B ∩ C).
please help
 
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  • #2
chica911 said:
The 2nd part is what I need help with. I am not sure how to prove (A ∩ C)- B = (A ∩ C) – (B ∩ C).
please help

The way to attack these is first, to break it into two parts, and second, for each part just read it left to right in English. Let me show you what I mean.

You have to prove two things:

(=>) If x is in (A ∩ C)- B then x is in (A ∩ C) – (B ∩ C) and

(<=) If x is in (A ∩ C) – (B ∩ C) then x is in (A ∩ C) - B.

Ok let's do the => direction. Suppose x is in (A ∩ C) - B. What does that mean in English? x is in A, and x is in C, but x is not in B.

Now on the right side, we must show that (A ∩ C) – (B ∩ C). We are given that x is in A, and we are given that x is in C; so x is in (A ∩ C). But we're given that x is not in B, so x is not in (B ∩ C). And that's what we had to show: that x is in (A ∩ C) – (B ∩ C). So the => is done.

Now you do the <= direction.
 
  • #3
chica911 said:
1. Let A , B and C be sets. Prove that (A-B) ∩ C = (A ∩ C)- B = (A ∩ C) – (B ∩ C).


3. Proof:
1st part: Let A, B and C be sets where (A-B) ∩ C. Let X be a particular, but arbitrary element of C. Since C and (A-B) ∩, X € (A-B) and X € C. Therefore, X € A but X ∉ B. Since X is an element of A and C, A ∩ C and since X ∉ B, (A ∩ C)-B. Therefore (A-B) ∩ C=(A ∩ C)-B.


if you pick an arbitrary element of C, you cannot claim it must lie in A-B as well, just because A-B and C intersect. horse first, THEN cart. choose x arbitrary in (A-B)∩C (and keep in mind there is no guarantee this set is not empty, because it might be).

what you have done above could be turned into an argument that (A-B)∩C is contained in (A∩C) - B, but the reverse containment needs to be shown as well.


2nd part: Let A, B and C be sets where (A ∩ C)- B.
3rd part: Suppose (A-B) ∩ C=(A ∩ C)-B and (A ∩ C)- B = (A ∩ C) – (B ∩ C) By the transitive property (A-B) ∩ C=(A ∩ C) – (B ∩ C) and therefore (A-B) ∩ C = (A ∩ C)- B = (A ∩ C) – (B ∩ C).

The 2nd part is what I need help with. I am not sure how to prove (A ∩ C)- B = (A ∩ C) – (B ∩ C).
please help

you are supposing what you wish to prove is true. that's a no-no.

what you want to do, is show (A∩C) - B is contained in (A∩C) - (B∩C) and vice versa.

so you suppose x is (A∩C) - B. this then, means x is in both A and C (since x is in A∩C), but not in B. well, if x is not in B, it cannot lie in any subset of B. so...

the other direction will be harder, because you are taking a "bigger bite" out of A∩C when you remove all the elements of B, then just removing all the elements of B∩C. so you need to show that A∩C doesn't contain any elements of B, EXCEPT those elements of B that already intersect C.
 

FAQ: How Can You Prove Set Equality for (A-B) ∩ C and (A ∩ C) - B?

What is an intersection in mathematics?

An intersection in mathematics is the set of elements that are shared by two or more sets. It is denoted by the symbol ∩ and can also be thought of as the common elements between the sets.

How do you prove that two sets have an intersection?

To prove that two sets have an intersection, you must show that there exists at least one element that is present in both sets. This can be done by using an element of the intersection as a common element in both sets and using set notation to express this relationship.

Can you give an example of a proof involving intersection?

Yes, for example, if we have the sets A = {1, 2, 3} and B = {2, 3, 4}, we can prove that they have an intersection by showing that the element 2 is present in both sets. This can be written as A ∩ B = {2}.

What is the relationship between union and intersection in set theory?

The union of two sets is the combination of all elements from both sets, while the intersection is the common elements between the two sets. In set theory, the union and intersection are complementary operations, meaning that the elements in the union are all the elements that are either in one set or the other, while the elements in the intersection are only those that are present in both sets.

How is intersection used in real life?

Intersection is used in real life in various fields, such as economics, statistics, and computer science. In economics, it is used to analyze overlapping markets and consumer preferences. In statistics, it is used to find common elements in data sets and to compare groups. In computer science, it is used in database queries to find data that satisfies multiple conditions.

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