CBSE Class 12 Computer Science: Chapter 6 Boolean Algebra NCERT Solutions

Solution: The OR operator (A + B) can be expressed using AND and NOT operators by applying De Morgan's laws. De Morgan's laws state that $(X\; \backslash cdot\; Y)\text{'}\; =\; X\text{'}\; +\; Y\text{'}$ and $(X\; +\; Y)\text{'}\; =\; X\text{'}\; \backslash cdot\; Y\text{'}$.

Starting with the expression for NOT (A + B), which is $(A\; +\; B)\text{'}$, we can apply the second law to get $A\text{'}\; \backslash cdot\; B\text{'}$. Now, if we take the complement of this result, we get $(A\text{'}\; \backslash cdot\; B\text{'})\text{'}$. Applying the rule that $(X\text{'})\text{'}\; =\; X$, we find that $(A\text{'}\; \backslash cdot\; B\text{'})\text{'}\; =\; A\; +\; B$. Therefore, the OR operator can be expressed as $(A\text{'}\; \backslash cdot\; B\text{'})\text{'}$ using only AND and NOT operators.

Solution:

1. The reduction $(bc)\text{'}\; +\; bc\; =\; 1$ uses the Complementary Law. This law states that for any Boolean variable x, $x\; +\; x\text{'}\; =\; 1$. In this case, $x$ is represented by $(bc)$ and $x\text{'}$ by $(bc)\text{'}$.
2. The reduction $xyz\; +\; zx\; =\; xz$ uses the Absorption Law. The absorption law can be stated in two forms: $x\; +\; xy\; =\; x$ and $x(x\; +\; y)\; =\; x$. In this specific case, we can factor out $xz$ from both terms: $xz(y\; +\; 1)$. Since $y\; +\; 1\; =\; 1$ (Identity Law), the expression simplifies to $xz(1)\; =\; xz$. This demonstrates the principle of absorption where the term $y$ is absorbed.

Solution: The gates known as universal gates are the NAND (Not AND) gate and the NOR (Not OR) gate.

They are called universal gates because any other basic logic gate (AND, OR, NOT) or any complex logic circuit can be constructed using only NAND gates or only NOR gates. This makes them highly versatile and cost-effective in digital circuit design.

Solution: The Associative Law in Boolean Algebra states that the grouping of variables does not affect the result of the operation. It applies to both OR and AND operations.

For OR operation: $(A\; +\; B)\; +\; C\; =\; A\; +\; (B\; +\; C)$

For AND operation: $(A\; \backslash cdot\; B)\; \backslash cdot\; C\; =\; A\; \backslash cdot\; (B\; \backslash cdot\; C)$

Verification using Truth Table for OR operation:

We need to show that the columns for $(A\; +\; B)\; +\; C$ and $A\; +\; (B\; +\; C)$ are identical.

A	B	C	$A\; +\; B$	$(A\; +\; B)\; +\; C$	$B\; +\; C$	$A\; +\; (B\; +\; C)$
0	0	0	0	0	0	0
0	0	1	0	1	1	1
0	1	0	1	1	1	1
0	1	1	1	1	1	1
1	0	0	1	1	0	1
1	0	1	1	1	1	1
1	1	0	1	1	1	1
1	1	1	1	1	1	1

As the columns for $(A\; +\; B)\; +\; C$ and $A\; +\; (B\; +\; C)$ are identical, the Associative Law for OR is verified.

Verification using Truth Table for AND operation:

Similarly, we can construct a truth table for $(A\; \backslash cdot\; B)\; \backslash cdot\; C\; =\; A\; \backslash cdot\; (B\; \backslash cdot\; C)$. The columns for $(A\; \backslash cdot\; B)\; \backslash cdot\; C$ and $A\; \backslash cdot\; (B\; \backslash cdot\; C)$ will be identical, thus verifying the Associative Law for AND.

A	B	C	$A\; \backslash cdot\; B$	$(A\; \backslash cdot\; B)\; \backslash cdot\; C$	$B\; \backslash cdot\; C$	$A\; \backslash cdot\; (B\; \backslash cdot\; C)$
0	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	1	1	0	0	1	0
1	0	0	0	0	0	0
1	0	1	0	0	0	0
1	1	0	1	0	0	0
1	1	1	1	1	1	1

The identical columns confirm that $(A\; \backslash cdot\; B)\; \backslash cdot\; C\; =\; A\; \backslash cdot\; (B\; \backslash cdot\; C)$.

Solution: The Boolean expression given is simply 'B'. This means the output of the logic circuit should be the same as the input variable B, regardless of the values of other variables (if any were present).

To draw the equivalent logic circuit, we simply need a wire that carries the input signal B directly to the output. This can be represented as a buffer, or more commonly, just by showing the input line B connected to the output point. No logic gate is required for a single variable expression like this, as it represents a direct pass-through of the signal.

Circuit Diagram:

Input: B

Output: B

(Imagine a simple line connecting the input 'B' to the output. No gate symbol is needed.)

Solution: The Principle of Duality is a fundamental concept in Boolean Algebra that allows us to derive new theorems and identities from existing ones.

Definition: The principle states that for any given Boolean expression or theorem, its dual expression is also true. The dual expression is obtained by performing the following transformations on the original expression:

Replace every OR operation (+) with an AND operation (.).

Replace every AND operation (.) with an OR operation (+).

Replace every constant 0 with 1.

Replace every constant 1 with 0.

For example, if the Identity Law A + 0 = A is true, its dual A \cdot 1 = A is also true.

Importance: The principle of duality is important because:

Simplification: It helps in simplifying complex Boolean expressions by providing a systematic way to derive related, often simpler, identities.

Theorem Derivation: It allows us to easily derive new theorems and properties of Boolean Algebra from known ones, reducing the effort required to establish all possible rules.

Completeness: It contributes to the completeness of the algebra by ensuring that for every valid statement, its dual is also valid, reflecting the symmetric nature of Boolean operations.

Essentially, it provides a shortcut for discovering and proving many properties of Boolean algebra.