Let A, B and C be sets. Prove that

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In summary, if A is a subset of the union of sets B and C, and the intersection of A and B is empty, then A is also a subset of C.
  • #1
iHeartof12
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*Sorry wrong section*
Let A, B and C be sets.
Prove that if A[itex]\subseteq[/itex]B[itex]\cup[/itex]C and A[itex]\cap[/itex]B=∅, then A[itex]\subseteq[/itex]C.

My attempted solution:
Assume A[itex]\subseteq[/itex]B[itex]\cup[/itex]C and A[itex]\cap[/itex]B=∅.
Then [itex]\vee[/itex]x (x[itex]\in[/itex]A[itex]\rightarrow[/itex]x[itex]\in[/itex]B[itex]\cup[/itex]x[itex]\in[/itex]c).

I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.
 
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  • #2
you almost have it.

for all x in A and A is a subset of B U C then x is in B U C
if x is in B U C then x is in B or x is in C

now just describe x with respect to the A [itex]\bigcap[/itex] B = ∅
 
  • #3
Another perspective: you can maybe rigorize it by saying:

a in B or a in C , but a not in B , so we must have a in C:

Basically, as you rightly concluded, a must be in one

of B or C, but ,by assumption/construction, a is not in B,

so a must be in C.
 

What does it mean to "Let A, B and C be sets"?

When we say "Let A, B and C be sets," we are defining A, B, and C as collections of objects or elements. These sets can contain any type of element such as numbers, letters, or even other sets. By declaring them as sets, we are establishing that they have certain properties and characteristics that we can use in a proof or argument.

What is a proof and why is it important?

A proof is a logical argument that uses established rules and principles to show that a statement or proposition is true. In the context of "Let A, B and C be sets," a proof is used to demonstrate that the given statement is valid and can be supported by evidence. Proofs are important in mathematics and science because they provide a way to verify the accuracy and validity of a statement or theory.

How do we prove that A, B, and C are sets?

In order to prove that A, B, and C are sets, we need to demonstrate that they follow the properties and characteristics of a set. This includes having well-defined elements, being able to determine if an element belongs to the set, and being able to perform set operations such as union, intersection, and complement. We can also use visual representations, such as Venn diagrams, to show the relationships between the sets.

What are the different techniques for proving statements about sets?

There are several techniques that can be used to prove statements about sets. These include direct proof, proof by contradiction, proof by induction, and proof by contrapositive. In a direct proof, we use logical steps to show that a statement is true. In a proof by contradiction, we assume the opposite of the statement and show that it leads to a contradiction. Proof by induction involves proving a statement for a base case and then showing that it holds for all other cases. In proof by contrapositive, we prove the logically equivalent statement by negating both the hypothesis and the conclusion.

Can we use examples to prove statements about sets?

Yes, examples can be used to support or illustrate a statement about sets, but they cannot be used as a proof. A proof requires a logical argument and cannot be based solely on examples. However, examples can help to provide a better understanding of the statement and can be useful in constructing a proof.

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