Subject - Theory of Computing 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: i. (f(x)) ii. -(f(x)) iii. f(x) + g(x) iv. f(x) - g(x) v. f(x)g(x) vi. f(x)/g(x) vii. f(x)g(x) viii. 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.

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
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Chapter1: Introduction
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Subject - Theory of Computing

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:
i. (f(x))
ii. -(f(x))
iii. f(x) + g(x)
iv. f(x) - g(x)
v. f(x)g(x)
vi. f(x)/g(x)
vii. f(x)g(x)
viii. 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.
2. 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= zn and y=zm for some integers n and m.

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