Give a Reason for Each Step in the Proof That $$x = x + 0 = x + ( x \bar { x } ) = ( x + x ) ( x + \bar { x } ) = ( x + x ) \cdot 1 = 1 \cdot ( x + x ) = x + x$$

question 14

Essay

Give a reason for each step in the proof that x + x = x is true in Boolean algebras. Your reasons should come from the following: associative laws for addition and multiplication, commutative laws for addition and multiplication, distributive law for multiplication over addition and distributive law for addition over multiplication, identity laws, unit property, and zero property. $$x = x + 0 = x + ( x \bar { x } ) = ( x + x ) ( x + \bar { x } ) = ( x + x ) \cdot 1 = 1 \cdot ( x + x ) = x + x$$

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Definitions:

Quasilinear Preferences

Preferences where utility is linear in one argument, usually money, allowing for easy analysis of changes in wealth while other goods are evaluated non-linearly.

Homothetic

Pertaining to a class of production functions or utility functions where equal proportionate changes in inputs result in equal proportionate changes in output.

Utility Function

An economic tool used to encode a consumer's preference orderings over a set of alternatives into a real-valued function. (Duplicate rephrase)

Income

Funds obtained regularly as a result of employment or from returns on investments.