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Design grammars for the following languages:

1. The set of all strings of $0's$ and $1's$ such that every $0$ is immediately followed by at least one $1$.
2. The set of all strings of $0's$ and $1's$ that are palindromes; that is, the string reads the same backward as forward.
3. The set of all strings of $0's$ and $1's$ with an equal number of $0's$ and $1's$.
4. The set of all strings of $0's$ and $1's$ with an unequal number of $0's$ and $1's$.
5. The set of all strings of $0's$ and $1's$ in which $011$ does not appear as a substring.
6.  The set of all strings of $0's$ and $1's$ of the form $xy$, where $x\neq y$ and $x$ and $y$ are of the same length.
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