Half Adders and Full Adders connection

Question

I found this strange thing upon my textbook

as an exercise to "comment by its truth table".

so I did the gargantuan truth table

But there's nothing to "comment", as I see.

The s2 = y0 reacts as a XOR or XNOR gate depending on x2, the c3 = y2 as an AND gate and the s3 = y1 seems entirely out of place.

Am Ι shortsighted at this?

Is there a connection in this design?

Answer 1

We shall start off with the inputs and outputs

• Inputs: X0, X1, X2, X3

• Outputs: Y0, Y1, Y2

This circuit basically sees how many inputs of X are HIGH and outputs as a number in binary through Y. So let me rearrange the truth table for better understanding.

X-inputs             -| Y-outputs
X3 X2 X1 X0          -| Y2 Y1 Y0
0  0  0  0  > 0 HIGH -| 0  0  0  > BINARY 0
0  0  0  1  > 1 HIGH -| 0  0  1  > BINARY 1
0  0  1  0  > 1 HIGH -| 0  0  1  > BINARY 1
0  1  0  0  > 1 HIGH -| 0  0  1  > BINARY 1
1  0  0  0  > 1 HIGH -| 0  0  1  > BINARY 1
0  0  1  1  > 2 HIGH -| 0  1  0  > BINARY 2
0  1  0  1  > 2 HIGH -| 0  1  0  > BINARY 2
1  0  0  1  > 2 HIGH -| 0  1  0  > BINARY 2
0  1  1  0  > 2 HIGH -| 0  1  0  > BINARY 2
1  0  1  0  > 2 HIGH -| 0  1  0  > BINARY 2
1  1  0  0  > 2 HIGH -| 0  1  0  > BINARY 2
0  1  1  1  > 3 HIGH -| 0  1  1  > BINARY 3
1  0  1  1  > 3 HIGH -| 0  1  1  > BINARY 3
1  1  0  1  > 3 HIGH -| 0  1  1  > BINARY 3
1  1  1  0  > 3 HIGH -| 0  1  1  > BINARY 3
1  1  1  1  > 4 HIGH -| 1  0  0  > BINARY 4

So mathematically

X0 + X1 + X2 + X3 = Y

Answer 2

If you had put y₂, y₁, and y₀ in the truth table you would have noticed that this circuit adds x₀, x₁, x₂, and x₃ and outputs the sum as y.