## Define the set of valid algebraic expressions. Please check the attached files for the question and answers

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Chapter 3, Problem 8P

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Introduction to Computer Theory (2nd Edition)

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Deﬁne the set of valid algebraic expressions ALEX as follows:

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Rule 1 All polynomials are in ALEX.

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

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(i) (f(x))

(ii) −(f(x))

(iii) f(x) + g(x)

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(iv) f(x) – g(x)

(v) f(x)g(x)

(vi) f(x)/g(x)

(vii) f(g(x))

(viii) f(g(x))

a) Show that (x + 2)-3x is in ALEX.

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b) Show that elementary calculus contains enough rules to prove the theorem that all algebraic

expressions can be differentiated.

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c) Is Rule 2 (viii) really necessary?

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Ph.D. in Mathematics

Step 1 of 4

Ryan

University of South…

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Consider the following rules:

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Rule 1: All polynomials are in ALEX.

Rule 2: If

and

are in ALEX, then so are:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Comment

Step 2 of 4

(a)

Consider the following polynomial:

Assume,

and

• Both expressions

.

and

• From Rule 2,

are polynomials. By Rule 1,

is in the form of

and

, where

are in ALEX.

and

.

Comment

Step 3 of 4

(b)

The statement of the theorem is that “All algebraic expressions can be differentiated”.

Algebraic expressions are the expressions containing constants, variable, elementary arithmetic

operations, factorial, integer and rational exponent and nth roots.

Constants:

• The expression containing constants are the expressions containing only numbers.

• These are the expressions containing the terms without variables.

• The value of constant expression never changes.

Ex: 3, 4, 5, -2 etc.

Variables:

• The expression containing variables are the expressions containing symbol for representing a

number.

• The value of variable can be changed when required.

Ex:

etc.

Elementary arithmetic expressions:

• The mathematical expressions containing variables, numbers and operations are elementary

arithmetic expressions.

• The arithmetic expressions are addition, subtraction, multiplication and division.

Ex:

etc.

Factorial:

• The factorial of an integer is the product of integers less than or equal to it.

• The multiplication between two constant values is also a constant which can be represented in

an expression.

Ex:

etc.

Integer and rational exponents:

• The integer exponent of an expression is the product of the expression speciﬁed by the integer

exponential value.

• The rational exponent of an expression is the product of the expression speciﬁed by the

fractional exponential value.

Ex:

etc.

n th roots:

• The nth roots of an expression is the number that should be multiplied n times itself to equal a

given value.

Ex:

As the algebraic expression for elementary calculus satisﬁes the enough rules required for

differentiating the theorem is satisﬁed.

Comment

Step 4 of 4

(c)

The rule

is not really necessary because, the expression is in the form of

with the value of input

is

. Here, the output of

is the input of

,

.

Comment

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