**Homework**** 1**

**(1 pt)**ALEX is the set of valid algebraic expressions recursively defined by:

Rule 1: All polynomials are in ALEX

Rule 2: If f(x) and g(x) are in ALEX, then so are:

- (f(x))
- -(f(x))

- f(x) + g(x)

- f(x) – g(x)
- f(x)g(x)
- f(x)/g(x)

- f(x)
^{g(x)} - f(g(x))

Assuming that the rules seen in class recursively defining polynomials can be used here to prove x+2 and 3x are polynomials, show that (x + 2)^{3x} is in ALEX.

**(1 pt)**Let x and y be two words (both different than the empty string) and xy is their concatenation. Show that if x, y and xy are all in PALINDROME, then there is a word z such that x= z^{n}and y=z^{m}for some integers n and m.

**(1 pt)**Let S={ab, bb} and T={ab, bb, bbb}. Show that S*≠T* but that S*⊂T*.

**(1 pt)**Write the regular expression for the language of all strings, over alphabet {a, b},that end in a double letter,__e. ending in aa or bb but not ab or ba__

**(1 pt)**Draw Deterministic Finite Automata to accept the following sets of strings, over the alphabet {0,1},that contain exactly four 0s (not necessarily consecutive zeros)

**Extra Credit: (1 pt)** Write the regular expression for the language of all strings, over alphabet {0, 1}, the set of all strings in which every pair of adjacent zeros appears before any pair of adjacent ones. Justify.