Solution: The OR operator ($A\; +\; B$) can be expressed using AND ($\backslash cdot$) and NOT ($\backslash prime$ or overbar) operators by applying De Morgan's laws. One way is to use the identity $A\; +\; B\; =\; (\backslash overline\{A\}\; \backslash cdot\; \backslash overline\{B\})\text{'}$. This means that the result of A OR B is the same as the result of NOT ( (NOT A) AND (NOT B) ). This is derived from De Morgan's second theorem, which states $(\backslash overline\{A\}\; \backslash cdot\; \backslash overline\{B\})\text{'}\; =\; A\; +\; B$.

Solution:

1. The reduction $(bc)\text{'}\; +\; bc\; =\; 1$ uses the Complementary Law. This law states that for any Boolean variable $x$, $x\; +\; x\text{'}\; =\; 1$. Here, $x$ can be considered as $bc$, so the expression is in the form $x\; +\; x\text{'}\; =\; 1$.
2. The reduction $xyz\; +\; zx\; =\; xz$ uses the Absorption Law. The Absorption Law states that $x\; +\; xy\; =\; x$ and $x(x+y)\; =\; x$. In this case, we can factor out $xz$ from the expression: $xyz\; +\; zx\; =\; zx(y\; +\; 1)$. Since $y\; +\; 1\; =\; 1$ (Identity Law), this simplifies to $zx(1)\; =\; zx$. Alternatively, recognizing $zx$ as a common term, the law $x\; +\; xy\; =\; x$ applies where $x\; =\; zx$ and $y\; =\; y$ (after rearranging terms to $zx\; +\; xyz$).

Solution: The gates known as universal gates are the NAND gate and the NOR gate. They are called universal because any other basic logic gate (AND, OR, NOT) or any complex combinatorial logic circuit can be constructed using only NAND gates or only NOR gates. This property 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:

$$(A\; +\; B)\; +\; C\; =\; A\; +\; (B\; +\; C)$$ (Associative Law for OR)

$$(A\; \backslash cdot\; B)\; \backslash cdot\; C\; =\; A\; \backslash cdot\; (B\; \backslash cdot\; C)$$ (Associative Law for AND)

Verification using Truth Table for OR: To verify $(A\; +\; B)\; +\; C\; =\; A\; +\; (B\; +\; C)$, we construct a truth table with all possible combinations of inputs A, B, and C:

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

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

Verification using Truth Table for AND: Similarly, we can verify $(A\; \backslash cdot\; B)\; \backslash cdot\; C\; =\; A\; \backslash cdot\; (B\; \backslash cdot\; C)$:

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 for $(A\; \backslash cdot\; B)\; \backslash cdot\; C$ and $A\; \backslash cdot\; (B\; \backslash cdot\; C)$ confirm the validity of the Associative Law for AND.

Solution: (Note: The specific Boolean expression for Question 5 was not provided in the source text. Assuming a hypothetical expression like $A\; \backslash cdot\; (B\; +\; C)$ to illustrate the process.)

To draw the equivalent logic circuit for a Boolean expression, we identify the operations and the variables. For the expression $A\; \backslash cdot\; (B\; +\; C)$:

1. First, the operation inside the parenthesis $B\; +\; C$ is performed. This requires an OR gate with inputs B and C.
2. Next, the result of the OR operation ($B\; +\; C$) is ANDed with variable A. This requires an AND gate whose inputs are A and the output of the OR gate ($B\; +\; C$).

The resulting circuit would consist of one OR gate and one AND gate.

(A visual representation would typically be included here showing the logic gates and connections.)

Solution: The Principle of Duality is a fundamental concept in Boolean Algebra. It states that for any given Boolean expression or equation, its dual expression or equation is also valid. The dual is obtained by performing the following transformations:

• Replace every OR operation (+) with an AND operation (·).
• Replace every AND operation (·) with an OR operation (+).
• Replace every Boolean constant 0 with 1.
• Replace every Boolean constant 1 with 0.

For example, the dual of $(A\; +\; B)\; \backslash cdot\; 1\; =\; A\; +\; B$ is $(A\; \backslash cdot\; B)\; +\; 0\; =\; A\; \backslash cdot\; B$. Importance in Boolean Algebra: The principle of duality is extremely important because:

• Simplification: It allows us to derive new theorems and identities from existing ones without needing separate proofs. If a statement is true, its dual is automatically true.
• Completeness: It helps in understanding the symmetry of Boolean operations and ensures that for every property, there is a corresponding dual property.
• Circuit Design: It aids in designing and analyzing digital circuits by providing a systematic way to transform expressions and understand their counterparts.

Essentially, it doubles the utility of any proven theorem or identity, making the study and application of Boolean Algebra more efficient.