Simplifying the Boolean Expression: (p ∧ ¬q) ∨ q ∨ (¬p ∧ q)

We will simplify the given expression step by step using Boolean algebra laws.

Step 1: Write the Expression

Expression: (p ∧ ¬q) ∨ q ∨ (¬p ∧ q)

Step 2: Apply the Distributive Law

Notice that q appears in multiple terms. We can factor q from the last two terms:

q ∨ (¬p ∧ q) = q ∧ (1 ∨ ¬p) [Since A ∨ (B ∧ A) = A ∧ (1 ∨ B)]

But it's simpler to use the identity: A ∨ (B ∧ A) = A

So, q ∨ (¬p ∧ q) = q

Thus, the expression becomes: (p ∧ ¬q) ∨ q

Step 3: Apply the Distributive Law Again

Now, we have (p ∧ ¬q) ∨ q. This can be rewritten using distribution:

(p ∨ q) ∧ (¬q ∨ q)

But ¬q ∨ q is always true (T), so:

(p ∨ q) ∧ T = p ∨ q

Step 4: Final Simplification

Therefore, the entire expression simplifies to p ∨ q.

Final Answer

The expression is equivalent to: p $\vee$ q

Related Topics

• Boolean Algebra
• Logical Equivalence
• Truth Tables
• De Morgan's Laws
• Propositional Logic

Important Formulae and Theory

Key Boolean Laws:

• Identity Laws: $A\vee 0=A$, $A\wedge 1=A$
• Domination Laws: $A\vee 1=1$, $A\wedge 0=0$
• Idempotent Laws: $A\vee A=A$, $A\wedge A=A$
• Complement Laws: $A\vee \mathrm{\neg A}=1$, $A\wedge \mathrm{\neg A}=0$
• Distributive Laws: $A\vee (B\wedge C)=(A\vee B)\wedge (A\vee C)$, $A\wedge (B\vee C)=(A\wedge B)\vee (A\wedge C)$
• Absorption Laws: $A\vee (A\wedge B)=A$,

Note: In this problem, we used the absorption law implicitly: $q\vee (\mathrm{\neg p}\wedge q)=q$ and the distributive law to simplify $(p\wedge \mathrm{\neg q})\vee q=(p\vee q)\wedge (\mathrm{\neg q}\vee q)=(p\vee q)\wedge 1=p\vee q$.