# Thread: What does this even mean? So confused...

1. ## What does this even mean? So confused...

I am totally stuck on this problem.

Suppose that $\displaystyle A$ is a subset of $\displaystyle V^{*}$, where $\displaystyle V$ is an alphabet. Prove or disprove each of these statements.

a) $\displaystyle A \subseteq A^2$
b) $\displaystyle \mbox{If} \ A = A^2, \mbox{then} \ \lambda \in A$
c) $\displaystyle A\{\lambda\} = A$
d) $\displaystyle (A^{*})^{*} = A^{*}$
What do all these symbols even mean i.e. *, lambda, etc. and how do I go about in solving these problems? Thank you.

2. Originally Posted by VENI
I am totally stuck on this problem.

Suppose that $\displaystyle A$ is a subset of $\displaystyle V^{*}$, where $\displaystyle V$ is an alphabet. Prove or disprove each of these statements.

a) $\displaystyle A \subseteq A^2$
b) $\displaystyle \mbox{If} \ A = A^2, \mbox{then} \ \lambda \in A$
c) $\displaystyle A\{\lambda\} = A$
d) $\displaystyle (A^{*})^{*} = A^{*}$
What do all these symbols even mean i.e. *, lambda, etc. and how do I go about in solving these problems? Thank you.
I am not sure. Doing a wiki search on "alphabet subset theory" I came up with this.

Free monoid - Wikipedia, the free encyclopedia

$\displaystyle V^{*}$ may mean an alphabet set minus the empty string.

$\displaystyle \lambda$ may be the empty string.

Guessing, $\displaystyle A^2$ may be the 2-tuples of elements of A.

With the above, I further conjucture weakly:

a) is false. $\displaystyle A \$ is not a 2-tuple.
b) is false. $\displaystyle A \$ is not a 2-tuple. $\displaystyle A = A^2 \$ does not result in $\displaystyle \ \ \lambda \in A$
c) not sure what $\displaystyle A\{\lambda\}$ implies.
d) is true. Removing the empty string twice from the set $\displaystyle A$ which already does not contain the empty string, is equivalent to removing the empty string once.

Supply more context on the source of this, for example: title of book, paragraph, section or chapter, or class topic under discussion.

3. Originally Posted by aleph1
I am not sure. Doing a wiki search on "alphabet subset theory" I came up with this.

Free monoid - Wikipedia, the free encyclopedia

$\displaystyle V^{*}$ may mean an alphabet set minus the empty string.

$\displaystyle \lambda$ may be the empty string.

Guessing, $\displaystyle A^2$ may be the 2-tuples of elements of A.

With the above, I further conjucture weakly:

a) is false. $\displaystyle A \$ is not a 2-tuple.
b) is false. $\displaystyle A \$ is not a 2-tuple. $\displaystyle A = A^2 \$ does not result in $\displaystyle \ \ \lambda \in A$
c) not sure what $\displaystyle A\{\lambda\}$ implies.
d) is true. Removing the empty string twice from the set $\displaystyle A$ which already does not contain the empty string, is equivalent to removing the empty string once.

Supply more context on the source of this, for example: title of book, paragraph, section or chapter, or class topic under discussion.
Thank you for your help. The problem comes from a hand out my professor gave me about finite-state automata.