Subset Question

  • #1

Homework Statement


I am trying to prove the absorption law
A U (A ∩ B) = A
I know that a way to prove this is to show that each is a subset of the other but I'm a little confused about one part in the process (below)

Homework Equations




The Attempt at a Solution


Let x∈A U (A ∩ B)
then x∈A or x∈(A ∩ B)
so because x∈A then we know that A U (A ∩ B) ⊆ A (I don't understand why this line is true.)

Why just because x∈A does it mean that A U (A ∩ B) ⊆ A is true? Any help is greatly appreciated.
 
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Answers and Replies

  • #2
haruspex
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then x∈A or x∈(A ∩ B)
so because x∈A then we know that A U (A ∩ B) ⊆ A
Arguably there's a step missing in there.
If x∈(A ∩ B) then x∈A , so either way x∈A U (A ∩ B) implies x∈A.
Thus you have shown that every element of A U (A ∩ B) is an element of A. Hence A U (A ∩ B) ⊆ A.
 
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  • #3
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so because x∈A then we know that A U (A ∩ B) ⊆ A (I don't understand why this line is true.)
This is true because each element in the subset ' A U (A ∩ B) ' must belong to A .
 
  • #4
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This is a logical argument. You are trying to show that if x is in ##A \cup ( A \cap B)##, then it is also in A, and if x is in A, then it is in ##A \cup ( A \cap B)##.
You have already shown the first part (edit) by the definition of the intersection: if x is in ##A \cup ( A \cap B)##, then it is also in A, which implies that ##A \cup ( A \cap B)\subseteq A ##,
Next, you need to show that ##A \subseteq A \cup ( A \cap B) ##. That should be simple enough by the definition of a union. So it looks like you are just about done.
 

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