MHB Prove relationship between sets

  • Thread starter Thread starter biocamme
  • Start date Start date
  • Tags Tags
    Relationship Sets
AI Thread Summary
The discussion focuses on proving the relationships between two sets A and B using set theory notation. The two key identities to prove are that the complement of the union of A and B equals the intersection of their complements, and the complement of the intersection of A and B equals the union of their complements. Participants suggest using Venn diagrams and the method of subset proof, which involves demonstrating that if an element belongs to one set, it must also belong to the other. The proof process is outlined, showing how to derive the relationships step by step. Overall, the conversation emphasizes the importance of clarity in notation and the logical structure of set proofs.
biocamme
Messages
1
Reaction score
0
For any two sets A and B prove:

(A∪B)^c=A^c∩B^c
(A∩B)^c=A^c∪B^c
 
Physics news on Phys.org
Hello biocamme and welcome to MHB! :D

I've edited the title of your thread to be more descriptive of the problem at hand. Also, we ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

A brief description of the notation you are using may be helpful to some.
 
biocamme said:
For any two sets A and B prove:

(A∪B)^c=A^c∩B^c
(A∩B)^c=A^c∪B^c

By using Venn diagram?
 
\text{For any two sets }A\text{ and }B.\:\text{ prove: }\; \begin{array}{cc} (A \cup B)^c\:=\:A^c \cap B^c \\ (A \cap B)^c \:=\:A^c \cup B^c \end{array}
By Venn diagrams? . Truth tables? . Other?

 
A standard method for proving two sets, X and Y, equal is to prove first that X\subseteq Y and then that Y\subseteq X. And to prove X\subseteq Y, start "if x\in X" and then use the properties of X and Y to conclude x\in Y.

Here, if x\in (A\cup B)^c x is not in A\cup B. So x is not in A and x is not in B. Since x is not in A then it is in A^c . Since x is not in B, then it is in B^c so x is in A^c\cap B^c

Now, do the other way- if x is in A^c\cap B^c then it is in both A^c and B^c so x is not in A and not in B. That is, x is not in A\cup B so is in (A\cup B)^c.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

Similar threads

Replies
18
Views
2K
Replies
2
Views
2K
Replies
5
Views
2K
Replies
11
Views
4K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
1
Views
2K
Back
Top