Question

You can upgrade your operating system only if you have a 32-bit processor running at 1 GHz or faster, at least 1 GB RAM, and 16 GB free hard disk space, or a 64- bit processor running at 2 GHz or faster, at least 2 GB RAM, and at least 32 GB free hard disk space. Express you answer in terms of u: “You can upgrade your operating system,” b32: “You have a 32-bit processor,” b64: “You have a 64-bit processor,” g1: “Your processor runs at 1 GHz or faster,” g2: “Your processor runs at 2 GHz or faster,” r1: “Your processor has at least 1 GB RAM,” r2: “Your processor has at least 2 GB RAM,” h16: “You have at least 16 GB free hard disk space,” and h32: “You have at least 32 GB free hard disk space.”

Short answer

Thelogical proposition of the given statement is determined as, $\mathbf{u}\mathbf{\to }\mathbf{\left(}\mathbf{\right(}{\mathbf{b}}_{\mathbf{32}}\mathbf{\wedge }\mathbf{g}\mathbf{\wedge }\mathbf{r}\mathbf{\wedge }{\mathbf{h}}_{\mathbf{16}}\mathbf{)}\mathbf{\vee }\mathbf{(}{\mathbf{b}}_{\mathbf{64}}\mathbf{\wedge }{\mathbf{g}}_{\mathbf{2}}\mathbf{\wedge }{\mathbf{r}}_{\mathbf{2}}\mathbf{\wedge }{\mathbf{h}}_{\mathbf{32}}\mathbf{\left)}\mathbf{\right)}$.

Step-by-step solution

Introduction to the Concept

"If-then" is a logical sign. It can be depict as $\mathbf{\to }$.

"or" is a logical sign. It can be depict as role="math" localid="1668083891892" $\mathbf{\vee }$.

"and" is a logical sign. It can be depict as$\mathbf{\wedge }$.

Solution Explanation

The given statements are,

u: You can upgrade your operating system

b32: You have a 32-bit processor

b64: You have a 64-bit processor

g1: Your processor runs at 1 GHz or faster

g2: Your processor runs at 2 GHz or faster

r1: Your processor has at least 1 GB RAM

r2: Your processor has at least 2 GB RAM

h16: You have at least 16 GB free hard disk space

h32: You have at least 32 GB free hard disk space

The given statement's propositional logical expression is

$\mathbf{u}\mathbf{\to }\mathbf{\left(}\mathbf{\right(}{\mathbf{b}}_{\mathbf{32}}\mathbf{\wedge }\mathbf{g}\mathbf{\wedge }\mathbf{r}\mathbf{\wedge }{\mathbf{h}}_{\mathbf{16}}\mathbf{)}\mathbf{\vee }\mathbf{(}{\mathbf{b}}_{\mathbf{64}}\mathbf{\wedge }{\mathbf{g}}_{\mathbf{2}}\mathbf{\wedge }{\mathbf{r}}_{\mathbf{2}}\mathbf{\wedge }{\mathbf{h}}_{\mathbf{32}\mathbf{\left)}\mathbf{\right)}}$.