Question

Match the following logical expression to its equivalent expression: $(\mathrm{\neg }(p\vee q))\vee (\mathrm{\neg }p\wedge q)$
(a) $p$
(b) $q$
(c) $\mathrm{\neg }p$
(d) $\mathrm{\neg }q$

Answer 1

Hint: We should be clear about the basic rules for simplifying logical expressions. The various laws we should know include Idempotence law, Commutative law, Associative law, Distributive law. We can simplify this question by following a few steps; first try expanding the negation using distributive law, remember the properties of ‘and ‘ and ‘or’ where ‘ $\wedge$ ’ stands for ‘and’, ‘ $\vee$ ’ stands for ‘or’. Try grouping based on the given brackets.

Complete step-by-step answer:
Observe the question carefully;
 $(\mathrm{\neg }(p\vee q))\vee (\mathrm{\neg }p\wedge q)$
Think about which all laws we can use to simplify this expression;
Start simplifying from the terms in the innermost bracket.
We see that there is a negation for the innermost bracket so we simplify in this way;
 $\Rightarrow (\mathrm{\neg }(p\vee q))\vee (\mathrm{\neg }p\wedge q)$
 $\Rightarrow (\mathrm{\neg }p\wedge \mathrm{\neg }q)\vee (\mathrm{\neg }p\wedge q)$ [ By De Morgan’s law]
 $\Rightarrow (\mathrm{\neg }p\wedge \mathrm{\neg }q)\vee (q\wedge \mathrm{\neg }p)$ [ By commutative law, the second bracket gets simplified ]
 $\Rightarrow (\mathrm{\neg }p\wedge (\mathrm{\neg }q\vee q)\wedge \mathrm{\neg }p)$ [ By associative law, we can group in this manner]
 $\Rightarrow (\mathrm{\neg }p\wedge \mathrm{\neg }p)$ [ We know that union of a term and its negation becomes $1$ and anything intersection $1$ remains as that term itself]
 $\Rightarrow \mathrm{\neg }p$ [ Intersection of a term with itself is the term itself , by Idempotent law]
Final answer is option (c) $\Rightarrow \mathrm{\neg }p$
So, the correct answer is “OPTION C”.

Note: Logical expressions follow a certain set of laws. We can list a few:
* Commutative law: This law says that there is no particular order we need to follow while solving logical expressions. It can be represented in this way:
 $\Rightarrow a\wedge b=b\wedge a$ or $\Rightarrow a\vee b=b\vee a$
* Associative law: This law says that order of grouping terms does not matter. Its representation is as follows:
 $\Rightarrow a\wedge (b\wedge c)=(a\wedge b)\wedge c$ or $\Rightarrow a\vee (b\vee c)=(a\vee b)\vee c$
* Distributive law: This is used when we have to use two operations together in an expression:
 $\Rightarrow a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)$ or $\Rightarrow a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)$
* De Morgan’s law: This is another form of a distributive law
 $\Rightarrow \mathrm{\neg }(a\vee b)=\mathrm{\neg }a\wedge \mathrm{\neg }b$ or $\Rightarrow \mathrm{\neg }(a\wedge b)=\mathrm{\neg }a\vee \mathrm{\neg }b$