Summary of Output Equations for
Common Combinational Logic Circuits

© 2003 by Charles C. Lin. All rights reserved.

This is a summary of output equations for some kinds of combinational logic circuits. This is done to make it easier to look it up and compare. Read the notes on each of the devices.

Output Equation for 4-1 MUX

This equation comes from the condensed truth table. 
  z = \c1\c0x0 +
      \c1c0x1 +
      c1\c0x2 +
      c1c0x3

Output Equation for 1-4 DeMUX

This equation comes from the condensed truth table. 
  z0 = \c1\c0x
  z1 = \c1c0x
  z2 =  c1\c0x
  z3 =  c1c0x

Output Equation for 2-4 Decoder

This equation comes from the condensed truth table. 
  z0 = \x1\x0
  z1 = \x1x0
  z2 =  x1\x0
  z3 =  x1x0

Output Equation for 2-4 Decoder with enable

This equation comes from the condensed truth table.

In this example, e is an active-high enable 

  z0 = \x1\x0e
  z1 = \x1x0e
  z2 =  x1\x0e
  z3 =  x1x0e
In this example, e is an active-low enable 
  z0 = \x1\x0\e
  z1 = \x1x0\e
  z2 =  x1\x0\e
  z3 =  x1x0\e

Output Equation for a 4-2 Regular Encoder

This is an encoder where exactly one input is 1 (rest of the inputs are 0). The output, z1z0 is in UB and encodes the input index that is 1. Thus, if x3 = 1, then the output is z1z0 = 11 since valueUB["11"] = 3 or equivalently reprUB[3ten] = "11". 

  z1 = x3 + x2
  z0 = x3 + x1

Output Equation for a 4-2 Priority Encoder

This is an encoder where at most one input is 1 (rest of the inputs are 0). In this case, the priority scheme is to say higher valued indexes have higher priority. Thus, x3 = 1 has the highest priority and x0 = 1 has the lowest. 

  z1 = x3 + x2
  z0 = x3 + \x2x1

Output Equation for Half Adder 

        sum = x XOR y
  carry out = xy

Output Equation for Full Adder 

        sum = x XOR y XOR cin
  carry out = xy + xcin + ycin
The carry out is 1 when any two of the three bits are 1. We can write each of the three product terms below and explain when their values become 1. Thus, the entire expression is 1 when any (or all) of the three product terms evaluate to 1. If only one of the three variables, x, y and cin are 1, then all three product terms are 0, and thus there's no carry. Clearly, if all values are 0, then so is the carry out.

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