Simplify F ·G ·H̅+F ·𝒢 ·H+F̅ ·G ·H With reference to Table 11.7: Reference F ·G ·H̅+F ·G ·H

step 1 Okay in this question we are asked that simplify the bullion equation which is f dot g plus hbar

Simplify F ·G ·H̅+F ·𝒢 ·H+F̅ ·G ·H With reference to Table...

Simplify $F \cdot G \cdot \bar{H}+F \cdot \mathcal{G} \cdot H+\bar{F} \cdot G \cdot H$ With reference to Table 11.7: Reference $$ \begin{aligned} F & \cdot G \cdot \bar{H}+F \cdot G \cdot H+\bar{F} \cdot G \cdot H \\ &=F \cdot G \cdot(\bar{H}+H)+\bar{F} \cdot G \cdot H \\ &=F \cdot G \cdot 1+\bar{F} \cdot G \cdot H \\ &=F \cdot G+\bar{F} \cdot G \cdot H \\ &=\boldsymbol{G} \cdot(\boldsymbol{F}+\overline{\boldsymbol{F}} \cdot \boldsymbol{H}) \end{aligned} $$ 5 10 12

Simplify
$F \cdot G \cdot \bar{H}+F \cdot \mathcal{G} \cdot H+\bar{F} \cdot G \cdot H$
With reference to Table 11.7:
Reference
$$
\begin{aligned}
F & \cdot G \cdot \bar{H}+F \cdot G \cdot H+\bar{F} \cdot G \cdot H \\
&=F \cdot G \cdot(\bar{H}+H)+\bar{F} \cdot G \cdot H \\
&=F \cdot G \cdot 1+\bar{F} \cdot G \cdot H \\
&=F \cdot G+\bar{F} \cdot G \cdot H \\
&=\boldsymbol{G} \cdot(\boldsymbol{F}+\overline{\boldsymbol{F}} \cdot \boldsymbol{H})
\end{aligned}
$$
5
10
12

Key Concepts

Boolean Algebra Simplification

This concept involves reducing complex Boolean expressions into simpler or more efficient forms while preserving logical equivalence. Techniques include the application of various Boolean laws to combine or eliminate terms, which is a common task in digital logic design and optimization.

Distributive Law

The distributive law allows one to factor common terms from Boolean expressions. It states that A · (B + C) = A · B + A · C, and is a fundamental tool in simplifying expressions by grouping similar terms together, enabling further reduction.

Complementarity Law

The complementarity law states that a variable ORed with its complement is equal to 1 (i.e., H + ¬H = 1). This law is crucial in simplification as it allows parts of an expression to be reduced to the identity element, thereby eliminating unnecessary complexity.

Identity Law

The identity law in Boolean algebra states that any expression ANDed with 1 remains unchanged (i.e., A · 1 = A). This principle is applied after simplifying parts of an expression to remove redundant factors and achieve a simpler, equivalent expression.

Factoring Common Terms

Factoring common terms involves identifying and extracting shared variables across multiple terms in a Boolean expression. This step leverages the distributive law to simplify the overall expression by reducing the number of distinct terms and making the expression more manageable for further simplification.

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