Sets and Quantifiers: Power Sets & Family Homework

• Terrell
In summary: OK, so it sounds like you are trying to solve this problem by "reverse engineering" it. That's a very inefficient way to solve problems. Instead, you should be using the definitions of the notation and the precise definitions of the logical concepts you are working with to figure out what are the necessary and sufficient conditions for a statement that is logically equivalent to the given statement.In this case, the given statement is already a symbolic representation of an English statement. So, one of the things you need to know is how to translate from English to symbolic representation. Do you know how to do that?OK, so it sounds like you are trying to solve this problem by "reverse engineering" it. That's a very
Terrell

Homework Statement

B ∈ {P (A) | A ∈ F}. where P(A) is the power set of A and F is Family

N/A

The Attempt at a Solution

My interpretation:
A: an element of the Family of sets. Hence, A is a set.
P(A): the set of all the possible unique subsets of A.
B: an element of the set of subsets of A, P(A). Thus, B⊆A.
∀x[x∈B → x∈A]

correct interpretation: ∃A ∈ F ∀x(x ∈ B ↔ ∀y(y ∈ x → y ∈ A))
i do not understand how that came to be and why my interpretation is wrong.

What is the exact problem? You didn't specify what you are having problems with.

Math_QED said:
What is the exact problem? You didn't specify what you are having problems with.
i do not understand why my interpretation is wrong. how should i read it? I've specified my problem in the attempt at a solution section of my post

Terrell said:
i do not understand why my interpretation is wrong.

So. what is the problem statement?

Is it:
Write a logical expression that is equivalent to ##B \in \{P(A): A \in F\}##
?

Terrell
Stephen Tashi said:
So. what is the problem statement?

Is it:

?
yes. that is where i am having trouble with. the steps in the book are as follows:
1) ∃A ∈ F(B = P (A))
2) ∀x(x ∈ B ↔ x ⊆ A)
3) ∃A ∈ F ∀x(x ∈ B ↔ ∀y(y ∈ x → y ∈ A))

I don't even understand how B=P(A). I thought B ∈ P(A) such that A ∈ F. My whole interpretation of B ∈ {P (A) | A ∈ F} is B is an element of the set of all subsets of A where A is an element of F. Thus, B is a proper subset of A and not simply any subset.
Now where in my interpretation did I went wrong? thank you.

Terrell said:

Homework Statement

B ∈ {P (A) | A ∈ F}. where P(A) is the power set of A and F is Family

You should quote the entire statement of the problem, exactly.

You've left your advisers guessing at what the "ground rules" are for solving it.

correct interpretation: ∃A ∈ F ∀x(x ∈ B ↔ ∀y(y ∈ x → y ∈ A))

My guess is that this is supposed to be a symbolic representation of a statement logically equivalent to:
"The set B is an element of the power set A and A is a element of the family of sets F"

For those words to be a statement (be it a true or false statement) the things that are declared to have a relation must exist or else there must be some convention about how notation indicating a relation is interpreted if one or both of the things happens not to exist.
yes. that is where i am having trouble with. the steps in the book are as follows:
1) ∃A ∈ F(B = P (A))
The meaning of that notation is unclear. It might mean "There exists an ##A## such that ##A \in F## and we define ##B## to be the power set of ##A##". That notation didn't bother to say that ##F## exists. Perhaps the ground rules are that ##F## is always assumed to exist. Apparently another assumption is that if set ##A## exists then set ##P(A)## exists.

2) ∀x(x ∈ B ↔ x ⊆ A)
That could be justified by the definition of ##B## as being ##P(A)##
3) ∃A ∈ F ∀x(x ∈ B ↔ ∀y(y ∈ x → y ∈ A))
If we recall ##B = P(A)## then that is a correct statement, but not knowing what the exact goal of the problem is, we can't say why that is "the answer".

i do not understand how that came to be and why my interpretation is wrong.
Not knowing the ground rules for the problem, I can't say why your interpretation is not the answer. As far as I can see your interpretation is a correct statement if we take for granted that certain things exist. However, I don't know what things we are allowed to assume exist and I don't know the exact conditions that "the answer" is required to satisfy.

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Stephen Tashi said:
Not knowing the ground rules for the problem, I can't say why your interpretation is not the answer. As far as I can see your interpretation is a correct statement if we take for granted that certain things exist. However, I don't know what things we are allowed to assume exist and I don't know the exact conditions that "the answer" is required to satisfy.
this was the ground rule: "Analyze the logical forms of the following statements." besides that we should know that P(A) is the power set of A and F means family set. It no longer elaborated what other sets were in F. All we should know is A is an element of F and B ∈ {P (A) | A ∈ F}.

Terrell said:
this was the ground rule: "Analyze the logical forms of the following statements." besides that we should know that P(A) is the power set of A and F means family set. It no longer elaborated what other sets were in F. All we should know is A is an element of F and B ∈ {P (A) | A ∈ F}.

If the exact statement of the problem is "Analyze the logical forms of the following statements" then we need to know how your text materials define a "logical form". Is there a definition for "logical form"? Are there some general instructions like "When we analyze the logical form of a statement we do the following..." ?

Terrell
Stephen Tashi said:
If the exact statement of the problem is "Analyze the logical forms of the following statements" then we need to know how your text materials define a "logical form". Is there a definition for "logical form"? Are there some general instructions like "When we analyze the logical form of a statement we do the following..." ?
it's using quantifiers in conditional statements, or statements, and statements, combining them all, etc...

Terrell said:
it's using quantifiers in conditional statements, or statements, and statements, combining them all, etc...

If the goal is to use as many quantifiers as possible then the "correct answer" beat yours in that respect (!).

Stephen Tashi said:
If the goal is to use as many quantifiers as possible then the "correct answer" beat yours in that respect (!).
then I really need help to know where i went wrong lol. thanks

i think i got it now. please do correct me if i misunderstood anything.

let x be an element of B,
Since x is an element of B and B is a subset of A then x must also be an element of A.
Since x is an element of A and by definition of subsets, x is also a subset of A which makes it an element of P(A).
Therefore, we can write ∀x(x ∈ B ↔ x ⊆ A).

1. What is a power set?

A power set is a set that contains all the possible subsets of a given set. For example, if the set A = {1, 2, 3}, then the power set of A would be {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}.

2. How do you calculate the size of a power set?

The size of a power set is equal to 2^n, where n is the number of elements in the original set. In other words, if a set has n elements, its power set will have 2^n elements.

3. What is a family of sets?

A family of sets is a collection of sets that are related in some way. For example, the family of all subsets of a given set is a family of sets.

4. What is the difference between universal and existential quantifiers?

The universal quantifier (∀) represents "for all" and is used to express that a statement is true for every element in a set. The existential quantifier (∃) represents "there exists" and is used to express that there is at least one element in a set for which the statement is true.

5. How can power sets and quantifiers be applied in real-world situations?

Power sets and quantifiers are commonly used in mathematics, computer science, and statistics. In real-world situations, they can be used to represent and analyze data, make predictions, and solve problems. For example, in statistics, quantifiers can be used to express the probability of an event occurring, while power sets can be used to represent all possible outcomes of an experiment.

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