# Prove A ∩ B ⊆ A ∪ B without Venn Diagrams

• Piffo
In summary, the notation A ∩ B represents the intersection of sets A and B, which is the set of elements that are common to both A and B. It is important to prove A ∩ B ⊆ A ∪ B without Venn Diagrams in order to have a more rigorous and formal mathematical proof, as well as to develop critical thinking and problem-solving skills. To prove A ∩ B ⊆ A ∪ B without Venn Diagrams, one can use the definition of set intersection and union, as well as logical deduction and mathematical properties such as the distributive law. An example of a proof for A ∩ B ⊆ A ∪ B without Venn Diagrams is by showing that if x is
Piffo

## Homework Statement

Prove that A intersection B is a subset of A union B without using Venn diagrams

## Homework Equations

No relevant equations really

## The Attempt at a Solution

What i am thinking is this:

A union B = A+B- A intersection B shows the intersection is within the union
So A intersection B = -A union B +A+B, which tells me the intersection is smaller than the union.

Can anybody give me some feedback please?

What you should be thinking about is this:
A is a subset of B if and only if for all $x\in A$, $x\in B$, the definition of "subset".

Start with "if x is in A intersect B then---", use whatever properties x must have because of that and wind up with "therefore x is in A union B".

Exactly what HallsofIvy said. Any time that you want to show something is a subset, you must show that all of the elements of the subset are also in the larger set. This one is particularly simple, because you only have two steps in the logic.

The same procedure can be used to show that two sets are equal. For example, if you want to show that the set A is equal to the set B, first show that all elements of A are in B, and also that all elements of B are in A. Then you can conclude that A and B are equal because they have the same elements.

## 1. What does the notation A ∩ B mean?

The notation A ∩ B represents the intersection of sets A and B, which is the set of elements that are common to both A and B.

## 2. Why is it important to prove A ∩ B ⊆ A ∪ B without Venn Diagrams?

Proving A ∩ B ⊆ A ∪ B without Venn Diagrams allows for a more rigorous and formal mathematical proof. It also helps to develop critical thinking and problem-solving skills.

## 3. How do you prove A ∩ B ⊆ A ∪ B without Venn Diagrams?

To prove A ∩ B ⊆ A ∪ B without Venn Diagrams, you can use the definition of set intersection and union, as well as logical deduction and mathematical properties such as the distributive law.

## 4. Can you provide an example of a proof for A ∩ B ⊆ A ∪ B without Venn Diagrams?

Yes, the proof can be shown as follows: Let x be an element in A ∩ B. By definition, x is in both A and B. Therefore, x is also in A ∪ B because it is in A. Hence, A ∩ B ⊆ A ∪ B.

## 5. What are some other methods for proving A ∩ B ⊆ A ∪ B besides using Venn Diagrams?

Other methods for proving A ∩ B ⊆ A ∪ B include using set identities, set builder notation, and mathematical induction. You can also use counterexample or proof by contradiction to show that A ∩ B ⊆ A ∪ B is not true.

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