Chapter B.1, Problem 11E

Solution

(a.)

The two logical equivalences discovered from the given pairs of statements.

It has been determined that the two logical equivalences discovered from the given pairs of statements, are:

  $\sim \left(p\vee q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\vee \sim q$

Given:

The following pairs of statements:

  $\sim \left(p\vee q\right)$ and $\sim p\vee \sim q$

  $\sim \left(p\vee q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\vee \sim q$

Concept used:

Logical statements are logically equivalent if their truth tables are identical.

Calculation:

The given pairs of statements are:

  $\sim \left(p\vee q\right)$ and $\sim p\vee \sim q$

  $\sim \left(p\vee q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\vee \sim q$

As discussed, logical statements are logically equivalent if their truth tables are identical.

As determined, the two logical equivalences discovered from the given pairs of statements, are:

  $\sim \left(p\vee q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\vee \sim q$

Conclusion:

It has been determined that the two logical equivalences discovered from the given pairs of statements, are:

  $\sim \left(p\vee q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\vee \sim q$

(b.)

An explanation of the analogy between the DeMorgan’s Laws for sets and those found in part (a).

It has been shown that DeMorgan’s Laws for sets corresponds (with respect to set analogy) to the logical equivalences obtained previously.

Given:

The following pairs of statements:

  $\sim \left(p\vee q\right)$ and $\sim p\vee \sim q$

  $\sim \left(p\vee q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\vee \sim q$

Concept used:

The logical symbols $\wedge$ , $\vee$ and $\sim$ corresponds to the set operations: set intersection ( $\cap$ ), set union ( $\cup$ ) and set complement ( ${}^C$ ) respectively.

Calculation:

As determined in part (a), the two logical equivalences discovered from the given pairs of statements, are:

  $\sim \left(p\vee q\right)$ and $\sim p\wedge \sim q$

  $\sim \left(p\wedge q\right)$ and $\sim p\vee \sim q$

Accordingly, $\sim \left(p\vee q\right)$ corresponds to ${\left(P\cup Q\right)}^C$ in set analogy, $\sim p\wedge \sim q$ corresponds to $P^C\cap Q^C$ in set analogy, $\sim \left(p\wedge q\right)$ corresponds to ${\left(P\cap Q\right)}^C$ in set analogy and $\sim p\vee \sim q$ corresponds to $P^C\cup Q^C$ in set analogy.

Now, the DeMorgan’s Laws for sets are given as:

  ${\left(P\cup Q\right)}^C=P^C\cap Q^C$ and ${\left(P\cap Q\right)}^C=P^C\cup Q^C$

It can be easily seen that DeMorgan’s Laws for sets, neatly corresponds to the logical equivalences obtained previously.

Conclusion:

It has been shown that DeMorgan’s Laws for sets corresponds (with respect to set analogy) to the logical equivalences obtained previously.