The following notation $| L_1L_2 |$ = $| L_1 |$ $| L_2 |$ denotes that the number of strings in the $L_1L_2$ concatenation is the same as the product of two numbers $| L_1 |$ and $| L_2 |$. If this statement is always true for any two languages, give formal arguments for proof it and if not, you need to show two languages $L_1$ and $L_2$ such that $| L_1L_2 |$ is not equal to$| L_1 |$ $| L_2 |$.
I tried to solve this by induction proof, but i don't see what is the step base , i'm confuse about what step base .I think in the proof is something like:
$\epsilon$ = void string
L1 = {$\epsilon$} then proof $|\{\epsilon\} L2 |$ = $|\{\epsilon\}|$ $| L_2 |$.
By definition, $\{\epsilon\} L_2 $ = $L_2$, so this is $|\{\epsilon\} L2 |$ = $|L_2|$
Now,it's easy to see |{$\epsilon$}| = 1 therefore $|\{\epsilon\}|$ $| L_2 |$ = 1 $|L_2|$ = $|L_2|$
Well, I don't know if I'm right, but if so, then how can I proceed with the inductive step. Thanks for the help.
The following notation $| L_1L_2 |$ = $| L_1 |$ $| L_2 |$ denotes that the number of strings in the $L_1L_2$ concatenation is the same as the product of two numbers $| L_1 |$ and $| L_2 |$. If this statement is always true for any two languages, give formal arguments for proof it and if not, you need to show two languages $L_1$ and $L_2$ such that $| L_1L_2 |$ is not equal to$| L_1 |$ $| L_2 |$.
I tried to solve this by induction proof, but i don't see what is the step base , i'm confuse about what step base .I think in the proof is something like:
L1 = {$\epsilon$} then proof $|\{\epsilon\} L2 |$ = $|\{\epsilon\}|$ $| L_2 |$.
By definition, $\{\epsilon\} L_2 $ = $L_2$, so this is $|\{\epsilon\} L2 |$ = $|L_2|$
Now,it's easy to see |{$\epsilon$}| = 1 therefore $|\{\epsilon\}|$ $| L_2 |$ = 1 $|L_2|$ = $|L_2|$
Well, I don't know if I'm right, but if so, then how can I proceed with the inductive step. Thanks for the help.
The following notation $| L_1L_2 |$ = $| L_1 |$ $| L_2 |$ denotes that the number of strings in the $L_1L_2$ concatenation is the same as the product of two numbers $| L_1 |$ and $| L_2 |$. If this statement is always true for any two languages, give formal arguments for proof it and if not, you need to show two languages $L_1$ and $L_2$ such that $| L_1L_2 |$ is not equal to$| L_1 |$ $| L_2 |$.
I tried to solve this by induction proof, but i don't see what is the step base , i'm confuse about what step base .I think in the proof is something like:
$\epsilon$ = void string
L1 = {$\epsilon$} then proof $|\{\epsilon\} L2 |$ = $|\{\epsilon\}|$ $| L_2 |$.
By definition, $\{\epsilon\} L_2 $ = $L_2$, so this is $|\{\epsilon\} L2 |$ = $|L_2|$
Now,it's easy to see |{$\epsilon$}| = 1 therefore $|\{\epsilon\}|$ $| L_2 |$ = 1 $|L_2|$ = $|L_2|$
Well, I don't know if I'm right, but if so, then how can I proceed with the inductive step. Thanks for the help.
How do i start this inductive proof on Languages? $| L_1L_2 |$ = $| L_1 |$ $| L_2 |$
The following notation $| L_1L_2 |$ = $| L_1 |$ $| L_2 |$ denotes that the number of strings in the $L_1L_2$ concatenation is the same as the product of two numbers $| L_1 |$ and $| L_2 |$. If this statement is always true for any two languages, give formal arguments for proof it and if not, you need to show two languages $L_1$ and $L_2$ such that $| L_1L_2 |$ is not equal to$| L_1 |$ $| L_2 |$.
I tried to solve this by induction proof, but i don't see what is the step base , i'm confuse about what step base .I think in the proof is something like:
L1 = {$\epsilon$} then proof $|\{\epsilon\} L2 |$ = $|\{\epsilon\}|$ $| L_2 |$.
By definition, $\{\epsilon\} L_2 $ = $L_2$, so this is $|\{\epsilon\} L2 |$ = $|L_2|$
Now,it's easy to see |{$\epsilon$}| = 1 therefore $|\{\epsilon\}|$ $| L_2 |$ = 1 $|L_2|$ = $|L_2|$
Well, I don't know if I'm right, but if so, then how can I proceed with the inductive step. Thanks for the help.