Q17Digital Electronics

A decoder is a highly specialized, multi-input, multi-output combinational logic circuit architected to convert binary-encoded information on its $n$ input lines directly into a maximum of $2^n$ unique, mutually exclusive active output lines. Fundamentally, a decoder acts as a hardware minterm generator. If a 3-to-8 decoder is supplied with three variables (A, B, C), its eight distinct output pins physically represent the eight possible canonical minterms ($m_0$ through $m_7$) of those specific variables. Only one output pin will ever be active (logic HIGH) at any given moment, directly corresponding to the exact binary value currently asserted on the inputs.

Implementing the Boolean Function

Because a decoder inherently generates every single possible minterm, it serves as an incredibly powerful, universal building block for constructing any arbitrary Boolean function expressed in Standard Sum of Products (SOP) form. The methodology simply requires connecting the required minterm output pins into a singular, multi-input OR gate.

The problem mandates the implementation of the function: $f(A,B,C)=\Sigma m(2,4,5,7)$.

Step 1: Decoder Setup We deploy a standard 3-to-8 decoder IC (like a 74LS138, though active-high is assumed here for simplicity). The three primary function variables $A$, $B$, and $C$ are physically connected to the decoder's select input pins ($S_2$, $S_1$, $S_0$ respectively, where A is the MSB). Ensure the decoder's Enable pin (if present) is permanently wired to the active state.

Step 2: Minterm Extraction and ORing The function explicitly demands that the final output $f$ goes HIGH if minterm 2 OR 4 OR 5 OR 7 is active. We physically wire a connection from the decoder's output pin $O_2$ (which represents $m_2=A^{\prime }B{C}^{\prime }$) into the first input of a 4-input OR gate. We wire output pin $O_4$ ($m_4=A{B}^{\prime }{C}^{\prime }$) to the second input of the OR gate. We wire output pin $O_5$ ($m_5=A{B}^{\prime }C$) to the third input of the OR gate. We wire output pin $O_7$ ($m_7=ABC$) to the fourth input of the OR gate.

Step 3: Final Output The output terminal of the 4-input OR gate perfectly represents the required Boolean function $f(A,B,C)$. If the binary inputs are 010 (decimal 2), output $O_2$ fires HIGH, triggering the OR gate, forcing $f$ HIGH. This elegant hardware topology entirely eliminates the need to physically construct complex networks of discrete AND gates and inverters.