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Found in: Page 35

### Discrete Mathematics and its Applications

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

# Find the dual of each of these compound propositions..a) ${p}{\wedge }{¬}{q}{\wedge }{¬}{r}$b) ${\left(}{p}{\wedge }{q}{\wedge }{r}{\right)}{\wedge }{s}$c) ${\left(}{p}{\vee }{F}{\right)}{\wedge }{\left(}{q}{\vee }{T}{\right)}$

a) ${p}{\vee }{¬}{q}{\vee }{¬}{r}$

b) ${\left(}{p}{\vee }{q}{\vee }{r}{\right)}{\vee }{s}$

c) ${\left(}{p}{\wedge }{T}{\right)}{\vee }{\left(}{q}{\wedge }{F}{\right)}$

See the step by step solution

## Step1:Definition of dual

The dual of a proposition which contains only the logical operators ${\vee }{,}{\wedge }{,}{¬}$, is the compound proposition obtained by replacing each ${\wedge }$ by ${\vee }$ , each ${\wedge }$ by ${\vee }$ , each ${\mathbf{T}}$ by , ${\mathbf{F}}$ and each ${\mathbf{F}}$ by ${\mathbf{T}}$

## Step 2: Dual of a)

Dual of ${p}{\wedge }{¬}{q}{\wedge }{¬}{r}$ is . ${p}{\vee }{¬}{q}{\vee }{¬}{r}$

## Step 2: Dual of b)

Dual of ${\left(}{p}{\wedge }{q}{\wedge }{r}{\right)}{\wedge }{s}$ is ${\left(}{p}{\vee }{q}{\vee }{r}{\right)}{\vee }{s}$

## Step 3: Dual of c)

Dual of ${\left(}{p}{\vee }{F}{\right)}{\wedge }{\left(}{q}{\vee }{T}{\right)}$ is ${\left(}{p}{\wedge }{T}{\right)}{\vee }{\left(}{q}{\wedge }{F}{\right)}$