Proving a Set: Venn Diagram Method for Homework Statement #1

[Kingyou123]

Homework Statement

Number 1

Homework Equations

I know I should use a Venn diagram.

The Attempt at a Solution

The statement says that x intersects y so therefore the statement equals x minus y. This Is my attempt at the solution, if you subtract y from the venn diagram you get this partially eaten cookie shape... So would that be enough to prove a it? My explanation would be that since what is left of x is clearly not a full set and some where y intersects x.

 

[andrewkirk]

A Venn diagram is not a proof.
I think they suggest you look at the Venn diagram to get an idea of why it's true, which should point you towards a formal proof, not for the diagram to serve as a proof.
An easy way to prove this is to write both sides in set-builder notation. Once you've done that, it's pretty easy to transform one side into the other using basic logical operations.

 

[Kingyou123]

A Venn diagram is not a proof.
I think they suggest you look at the Venn diagram to get an idea of why it's true, which should point you towards a formal proof, not for the diagram to serve as a proof.
An easy way to prove this is to write both sides in set-builder notation. Once you've done that, it's pretty easy to transform one side into the other using basic logical operations.

X ∩ Y(line above the Y) = {ℤ∈ℤ | ℤ∈x ∨ ℤ ∈Y} would this be the correct set builder notation.

 

[Mark44]

X ∩ Y(line above the Y) = {ℤ∈ℤ | ℤ∈x ∨ ℤ ∈Y} would this be the correct set builder notation.

No. For one thing, ℤ∈ℤ doesn't make any sense, because you haven't said what ℤ is, and a set is not an element of itself.

For another thing, ℤ∈x ∨ ℤ ∈Y, use lower case for elements of a set, and upper case for the sets themselves. I would write what you have as {$z \in U | z \in X ∨ z \in Y$}, where U is the universal set. But that isn't correct either. The symbol ∨ is the logical or -- for sets use U for union.

As mentioned, my revision of what you wrote is still wrong. It represents all elements that belong to X, together with all elements that belong to Y. That isn't what you want.

 

[Kingyou123]

No. For one thing, ℤ∈ℤ doesn't make any sense, because you haven't said what ℤ is, and a set is not an element of itself.

For another thing, ℤ∈x ∨ ℤ ∈Y, use lower case for elements of a set, and upper case for the sets themselves. I would write what you have as {$z \in U | z \in X ∨ z \in Y$}, where U is the universal set. But that isn't correct either. The symbol ∨ is the logical or -- for sets use U for union.

As mentioned, my revision of what you wrote is still wrong. It represents all elements that belong to X, together with all elements that belong to Y. That isn't what you want.

So for example I set A= X-Y and using the definition of set difference. A∈x and A∉Y. The definition of complement x ∉y implies x ∉Y(line above it). Then the definition of intersection makes it x∈X∩Y(line above it).
Then I would just to do the same thing with the other part correct?

 

[Mark44]

So for example I set A= X-Y

There's no need to bring another set into the mix.

and using the definition of set difference. A∈x and A∉Y.

This makes no sense. As you have defined A (which you really don't need), it's a set. Use x (lowercase) for set elements and X (uppercase) for set names.

The definition of complement x ∉y implies x ∉Y(line above it). Then the definition of intersection makes it x∈X∩Y(line above it).
Then I would just to do the same thing with the other part correct?

You want to show that if $x \in X \cap \overline{Y}$, then $x \in X - Y$. That shows that $X \cap \overline{Y} \subset X - Y$. Then you need to go the other way: if $x \in X - Y$, then $x \in X \cap \overline{Y}$, which will show that $X - Y \subset X \cap \overline{Y}$. Together, these two proofs show that the two sets are equal.

I used LaTeX for my set notation, with ## at the beginning of each, and the same at the end.
\overline{Y} to make $\overline{Y}$
\subset to make $\subset$
\cap to make $\cap$

 

[Kingyou123]

There's no need to bring another set into the mix.
This makes no sense. As you have defined A (which you really don't need), it's a set. Use x (lowercase) for set elements and X (uppercase) for set names.You want to show that if $x \in X \cap \overline{Y}$, then $x \in X - Y$. That shows that $X \cap \overline{Y} \subset X - Y$. Then you need to go the other way: if $x \in X - Y$, then $x \in X \cap \overline{Y}$, which will show that $X - Y \subset X \cap \overline{Y}$. Together, these two proofs show that the two sets are equal.

I used LaTeX for my set notation, with ## at the beginning of each, and the same at the end.
\overline{Y} to make $\overline{Y}$
\subset to make $\subset$
\cap to make $\cap$

That is a lot simpler way then what I did. And I'm sorry for my syntax errors