Negation of AUB: Constructing without Symbols | Homework Help

[mharten1]

Homework Statement

Construct the negation of each statement below without using the fol-
lowing symbols: ∩, ∪, ⊆, ⊇, ⊂, ⊃, \, △, and =.

x ∈ A U B

Homework Equations

The Attempt at a Solution

I don't even know where to start. I'm completely lost. =\

 

[STEMucator]

Homework Statement

Construct the negation of each statement below without using the fol-
lowing symbols: ∩, ∪, ⊆, ⊇, ⊂, ⊃, \, △, and =.

x ∈ A U B

Homework Equations

The Attempt at a Solution

I don't even know where to start. I'm completely lost. =\

Well, my guess is you want to negate the statement : x ∈ AUB.

This statement means that either x is in A or x is in B. How do you NEGATE this proposition?

 

[mharten1]

Well, my guess is you want to negate the statement : x ∈ AUB.

This statement means that either x is in A or x is in B. How do you NEGATE this proposition?

X is not in A or B? How do I symbolically write that? Because I can't use the union symbol. I initially thought that but it seemed too simple.

 

[STEMucator]

X is not in A or B? How do I symbolically write that? Because I can't use the union symbol. I initially thought that but it seemed too simple.

The negation operator is written as : \neg

 

[mharten1]

The negation operator is written as : \neg

Ok, that makes sense. But I can't use the union symbol, and that's where I'm really confused. Could I say ~(A v B)? ~A ^ ~B by demorgans law.

 

[STEMucator]

Ok, that makes sense. But I can't use the union symbol, and that's where I'm really confused.

Well you're not being very clear about what statements you're trying to negate. Are they phrases that you haven't posted or is it actually the statement : x∈AUB. If you're trying to negate x∈AUB, then you can take for granted that the question is ASKING you to negate that statement, you're simply just not allowed to use the union symbol in doing it.

 

[mharten1]

Well you're not being very clear about what statements you're trying to negate. Are they phrases that you haven't posted or is it actually the statement : x∈AUB. If you're trying to negate x∈AUB, then you can take for granted that the question is ASKING you to negate that statement, you're simply just not allowed to use the union symbol in doing it.

It's actually the statement x ∈ AUB.

 

[STEMucator]

It's actually the statement x ∈ AUB.

Like I've already stated before then, if x ∈ AUB then either x is in A or x is in B. How would you negate this proposition, aka write the opposite of what this means.

 

[mharten1]

Like I've already stated before then, if x ∈ AUB then either x is in A or x is in B. How would you negate this proposition, aka write the opposite of what this means.

I would do ~(A v B), which simplifies to ~A ^ ~B. But as far as I know A and B are sets so I can't use those symbols.

 

[STEMucator]

I would do ~(A v B), which simplifies to ~A ^ ~B. But as far as I know A and B are sets so I can't use those symbols.

\neg((x is in A)\vee(x is in B))

The fact that A and B are sets makes no difference. Think about what I wrote there.