Help with Set Theory

iowa

does anyone know anything about set theory?

(A/\B)' = A' U B'

/\ is an upside down U.


The first one is the set of all elements that are the total elements that are not in A and not in B.

The second one is the set of all elements not in A and not in B.

don't we need to know what the total set is in order to say if these are the same? Are we assuming the total set is encompassed by A+B+A'+B'?

Is it even possible for the total set to not be A+B+A'+B'?

kime

No, by definition A U B U A' U B' = the universe. You could simplify that to being either A U A', or B U B'.

More reasoning for that:

It's basically a definition thing. The definition of A' is U-A, where U is the universe you're looking at (the "total set", as you call it). It literally means "all elements in the 'total set' that are not in A." So if you have A U A', it literally means "all elements in A, and all elements not in A in the universe." Nothing is excluded.

Therefore, you can safely say that they are both the same.

Be very careful with the English, though. The reason we have an established set of symbols for this is because the English is ambiguous. For instance, what you said, "The second one is the set of all elements not in A and not in B," could be interpreted in two different ways (maybe more, doesn't matter). I know what you meant, but you should just be aware it may not be 100% clear, and, therefore, correct if you are just giving that as a description.

iowa

kime wrote: »

No, by definition A U B U A' U B' = the universe. You could simplify that to being either A U A', or B U B'.

More reasoning for that:

It's basically a definition thing. The definition of A' is U-A, where U is the universe you're looking at (the "total set", as you call it). It literally means "all elements in the 'total set' that are not in A." So if you have A U A', it literally means "all elements in A, and all elements not in A in the universe." Nothing is excluded.

Therefore, you can safely say that they are both the same.

Be very careful with the English, though. The reason we have an established set of symbols for this is because the English is ambiguous. For instance, what you said, "The second one is the set of all elements not in A and not in B," could be interpreted in two different ways (maybe more, doesn't matter). I know what you meant, but you should just be aware it may not be 100% clear, and, therefore, correct if you are just giving that as a description.

Ok Thanks for the clarification.

yeah it took a long time to even type out that post because English is hard in this regard.

tarnok

iowa wrote: »

yeah it took a long time to even type out that post because English is hard in this regard.

In relation to this, a couple of venn diagrams with some shading make the equivalence fairly clear. Human language is terrible for relating complex abstract ideas.

kime

tarnok wrote: »

iowa wrote: »

yeah it took a long time to even type out that post because English is hard in this regard.

In relation to this, a couple of venn diagrams with some shading make the equivalence fairly clear. Human language is terrible for relating complex abstract ideas.

One thing to note, for classes, professors tend to not prefer Venn diagrams for proofs, they're for your benefit. At least from my experience.

Sentient6

kime wrote: »

tarnok wrote: »

iowa wrote: »

yeah it took a long time to even type out that post because English is hard in this regard.

In relation to this, a couple of venn diagrams with some shading make the equivalence fairly clear. Human language is terrible for relating complex abstract ideas.

One thing to note, for classes, professors tend to not prefer Venn diagrams for proofs, they're for your benefit. At least from my experience.

Strictly speaking, it's not a preference; Venn Diagrams are not proofs. But like you said they can be very useful. The usual route for this kind of thing is to establish mutual containment, i.e. the left side contains the right side and the right side contains the left side.