Prove relationship between sets

In summary, (A∪B)^c=A^c∩B^c is proven by showing that any element x that is not in A∪B is also not in either A or B, and vice versa. Therefore, they are equal. Similarly, (A∩B)^c=A^c∪B^c is proven by showing that any element x that is not in A∩B is either not in A or not in B, and vice versa. This shows that the two sets are equal.
  • #1
biocamme
1
0
For any two sets A and B prove:

(A∪B)^c=A^c∩B^c
(A∩B)^c=A^c∪B^c
 
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  • #2
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.
 
  • #3
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?
 
  • #4
[tex]\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}[/tex]

By Venn diagrams? . Truth tables? . Other?

 
  • #5
A standard method for proving two sets, X and Y, equal is to prove first that [tex]X\subseteq Y[/tex] and then that [tex]Y\subseteq X[/tex]. And to prove [tex]X\subseteq Y[/tex], start "if [tex]x\in X[/tex]" and then use the properties of X and Y to conclude [tex]x\in Y[/tex].

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

Now, do the other way- if x is in [tex]A^c\cap B^c[/tex] then it is in both [tex]A^c[/tex] and [tex]B^c[/tex] so x is not in A and not in B. That is, x is not in [tex]A\cup B[/tex] so is in [tex](A\cup B)^c[/tex].
 

Related to Prove relationship between sets

1. How do you prove a relationship between sets?

In order to prove a relationship between sets, you must show that all elements in one set are also contained in the other set, and vice versa. This is known as the "subset" relationship. Additionally, you can prove a relationship between sets by showing that the two sets have the same number of elements, known as the "cardinality" relationship.

2. What is the difference between a subset and a cardinality relationship?

A subset relationship involves showing that all elements in one set are also contained in the other set, while a cardinality relationship involves showing that the two sets have the same number of elements. In other words, a subset relationship is a more strict condition than a cardinality relationship.

3. Can a set have a subset relationship with itself?

Yes, a set can have a subset relationship with itself. This is known as the "reflexive" property of subsets. For example, the set of all even numbers is a subset of the set of all integers, but it is also a subset of itself.

4. How can you disprove a relationship between sets?

To disprove a relationship between sets, you must find at least one element that is in one set but not in the other. This will show that the two sets are not equal or that one set is not a subset of the other.

5. Are there other types of relationships between sets besides subset and cardinality?

Yes, there are other types of relationships between sets. Some examples include the "disjoint" relationship, where two sets have no elements in common, and the "complement" relationship, where one set contains all elements that are not in the other set.

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