Q25Discrete Mathematics

The objective is to mathematically reduce the complex logical expression $(\neg p\to r)\wedge (p\leftrightarrow q)$ into its standard Disjunctive Normal Form (DNF). The DNF strictly requires the final expression to be a massive "Sum of Products" (a series of ANDed minterms connected by OR operators). Because the expression contains exactly three independent variables ($p,q,r$), the truth table requires $2^3=8$ rows to evaluate every possible Boolean state.

Step 1: Construct the 8-Row Truth Table

We must evaluate the intermediate components: $\neg p$, the implication $(\neg p\to r)$, and the biconditional $(p\leftrightarrow q)$. (Note: Implication $A\to B$ is False ONLY when $A$ is True and $B$ is False).

| $p$ | $q$ | $r$ | $\neg p$ | $(\neg p\to r)$ | $(p\leftrightarrow q)$ | Final Output: $(\neg p\to r)\wedge (p\leftrightarrow q)$ | |---|---|---|---|---|---|---| | T | T | T | F | T | T | T (True AND True) | | T | T | F | F | T | T | T (True AND True) | | T | F | T | F | T | F | F | | T | F | F | F | T | F | F | | F | T | T | T | T | F | F | | F | T | F | T | F (T$\to$F) | F | F | | F | F | T | T | T | T | T (True AND True) | | F | F | F | T | F (T$\to$F) | T | F |

Step 2: Isolate Valid Minterms

To construct the DNF, we scan the "Final Output" column and extract exclusively the specific rows that mathematically evaluated to True.

• Row 1: $p$ is True, $q$ is True, $r$ is True. - Minterm 1: $(p\wedge q\wedge r)$
• Row 2: $p$ is True, $q$ is True, $r$ is False. - Minterm 2: $(p\wedge q\wedge \neg r)$
• Row 7: $p$ is False, $q$ is False, $r$ is True. - Minterm 3: $(\neg p\wedge \neg q\wedge r)$

Step 3: Disjunctive Synthesis

The final mathematical step is to chain the extracted minterms together using the logical OR operator (disjunction). The definitive Disjunctive Normal Form is: DNF = $(p\wedge q\wedge r)\vee (p\wedge q\wedge \neg r)\vee (\neg p\wedge \neg q\wedge r)$

(Note: While this is a valid DNF, it could be further algebraically simplified to $(p\wedge q)\vee (\neg p\wedge \neg q\wedge r)$ if the problem asked for the absolute minimal DNF, but standard DNF strictly requires listing the canonical minterms as extracted).