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Found in: Page 22

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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.”

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

See the 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,

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

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