Question

The following sentence is taken from the specification of a telephone system: “If the directory database is opened, then the monitor is put in a closed state, if the system is not in its initial state.” This specification is hard to understand because it involves two conditional statements. Find an equivalent, easier-to-understand specification that involves disjunctions and negations but not conditional statements.

Short answer

For finding equivalent, easier-to-understand specification that involves disjunctions and negations but not conditional statements, write the given statement in symbolic form and find the equivalence statement using disjunctions and neagtions.

Step-by-step solution

Definition of logically equivalent propositions  

Any two propositions are said to belogically equivalentif they have same truth value for any combination of truth values ofpand q.

To find an equivalent statement that involves disjunctions and negations.

The given statement is taken as “If the directory database is opened and if the system is not in its initial state, then the monitor is put in a closed state.”

Express the given statement is symbolic form.

Take three input variablesp,qandr.

The symbolic form of statement is $\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{\neg }\mathbf{r}\mathbf{)}\mathbf{\to }\mathbf{q}$.

Follow logical equivalence law.

$\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{\neg }\mathbf{r}\mathbf{)}\mathbf{\to }\mathbf{q}\mathbf{=}\mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{\neg }\mathbf{r}\mathbf{)}\mathbf{\vee }\mathbf{q}$

Follow De morgan’s law

$\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{\neg }\mathbf{r}\mathbf{)}\mathbf{\to }\mathbf{q}\mathbf{=}\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{(}\mathbf{\neg }\mathbf{r}\mathbf{\left)}\mathbf{\right)}\mathbf{\vee }\mathbf{q}$

Use double negation law

$\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{\neg }\mathbf{r}\mathbf{)}\mathbf{\to }\mathbf{q}\mathbf{=}\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{r}\mathbf{)}\mathbf{\vee }\mathbf{q}$

The above equivalent statement has only disjunction and negations.

Therefore, the required statement is “The directory database is not opened or the system is in initial state, or the monitor is in closed state.”