Proof involving sets. NEED HELP

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In summary, the proof implies that if A U B = A, then B is a subset of A. However, if A U B ≠ A, then B is not a subset of A.
  • #1
arpitm08
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Proof involving sets. NEED HELP!

Homework Statement



Prove directly "If A U B = A, then B is a subset of A." and also provide a proof by contrapositive of its converse.


2. The attempt at a solution

Here is what i did, but I don't know if it is right or not,

Direct Proof: Assume A U B = A, then x ∈ (A U B) and x ∈ A. So it follows that B ∈ A = B is a subset of A.
Contrapositive of Converse Proof: Assume that A U B ≠ A, then x ∈ (A U B) and x ∉ A. Then, B ∉ A and so B is not a subset of A.

I don't think this is right. Could someone help me out please??
 
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  • #2


Direct Proof: Assume A U B=A as you have. Then to show a set B is a subset of a set A the standard technique is to let x be in B, then show it is also in A. Notice if x is in B then it is clearly also in A U B and the conclusion follows from your initial assumption.

Converse: If B is a subset of A then A U B=A
Contrapositive of Converse: If A U B /neq A then B is not a subset of A

Notice A U B /neq A but A is clearly a subset of A U B. So what's left to make that not equals is A U B is not a subset of A. That gives that there's an element in A U B that is not in A. Go from there.
 
  • #3


How about now...

Directly -
Assume A U B = A, then x ∈ B. Then x ∈ (A U B), and since A U B = A, x ∈ A. So B is a subset of A.

Contrapositive of Converse -
Assume that A U B ≠ A. Since A is a subset of A U B, there must be an x ∈ (A U B), such that x ∉ A, since A U B ≠ A. This means that there is a y ∈ B, such that y ∉ A. So B is not a subset of A.

Is that a complete proof??
 
  • #4


in contrapositive of converse

though its implied, I think you should change it to A is a proper subset of AUB
 
  • #5


Allright. Thanks! =D
 
  • #6


Looks good.
 

1. What is a set in mathematics?

A set is a collection of distinct objects, called elements, that are grouped together based on a specific criteria or property. Sets are commonly used in mathematics to represent and organize data.

2. How do you prove that two sets are equal?

To prove that two sets are equal, you need to show that they have the same elements. This can be done by using the element method, where you show that every element in one set is also in the other set, and vice versa. Another method is the subset method, where you show that each element in one set is a subset of the other set and vice versa.

3. What is the difference between a proper subset and a subset?

A subset is a set that contains all the elements of another set. A proper subset, on the other hand, is a subset that contains at least one element that is not in the original set. In other words, a proper subset is a subset that is smaller than the original set.

4. How do you prove that two sets are disjoint?

To prove that two sets are disjoint, you need to show that they have no elements in common. This can be done by using the element method, where you show that there is no element that is present in both sets. Another method is the intersection method, where you show that the intersection of the two sets is an empty set.

5. What is the difference between a union and an intersection of sets?

The union of two sets is a new set that contains all the elements from both sets. The intersection of two sets, on the other hand, is a new set that contains only the elements that are common to both sets. In other words, the union combines the elements from both sets, while the intersection finds the overlapping elements between the two sets.

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