# Proof A U (A ∩ B) ⊆ A: Understanding x∈A

• dirtybiscuit
In summary, by showing that each element of A U (A ∩ B) is an element of A, you have proved that A U (A ∩ B) ⊆ A.

## 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)

## 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.

Maged Saeed
dirtybiscuit said:
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.

Maged Saeed
dirtybiscuit said:
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 .

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.

## What does the expression "Proof A U (A ∩ B) ⊆ A" mean?

This expression means that any element x that is in both sets A and B is also in set A.

## How is this expression related to set theory?

This expression is related to set theory because it is using the symbols and concepts of sets to represent the relationship between the sets A and B.

## Why is this expression important in scientific research?

This expression is important in scientific research because it helps to define and understand the relationship between two sets of data or variables. It can also be used to make predictions and test hypotheses.

## Can you provide an example of how this expression is used in a scientific study?

Sure, for example, in a study on the effects of exercise on weight loss, the set A could represent the group of participants who exercised, and the set B could represent the group of participants who followed a specific diet. The expression "Proof A U (A ∩ B) ⊆ A" would mean that any participant who followed both the exercise and diet would also be included in the group who just exercised.

## How can understanding this expression benefit my research?

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