Half Adder and Two Function, A Contest Questions?

I ran into a 2013 contest question on computer-science filed.

What is the following True about F and G function. (The output of Decoder is zero when disabled).

I) Is equal.

II) Complement to each other.

III) the 1's in F is more than 1's in G.

IV) the 0's in F is more than 0's in G.

Answer Sheet Select (1), But I think (2) is Correct. Who can verify me and hint how we can correctly solve such a circuit?

Assuming the decoder is active, the Truth table is as follows :

XZ    Dec_Out    HA1        HA2    a b cin    F G
Sum | C    Sum | C
00   1000      1     0    0     0  1 1 0      0 1
01   0100      1     0    0     0  1 1 0      0 1
10   0010      0     0    1     0  0 1 1      0 1
11   0001      0     0    1     0  0 1 1      0 1


The truth table is easy to construct :

• First determine the output of decoder corresponding to various inputs
• Next apply D0 and D1 as inputs to HA1 and D2 and D3 as inputs to HA2
• Compute their outputs and apply them to FA

Now since it was a competition problem, there has to be a short cut (which I was not able to figure out initially as well) instead of solving the truth table :P

• Two outputs of a decoder cant be simultaneously high. This means that the sum of both half adders cant be one simultaneously. Infact, output of exactly one HA HAS to be 1 always, and also that both have to give a carry of zero.
• The Xnor will always give a 1, since both carry are always zero.
• If 2 of the inputs to the Full adder are always one, the sum ie F will always be 0, while the carry ie G will always be 1.

Similar arguments hold when decoder is inactive, but this time output of both HA will be 0.

• With outputs of both half adders as zero, the XNOR is still high while A and Cin are zero
• So the input of FA consists of exactly one, which means sum ie F will always be 1 while carry ie G will always be 0.

In either case, F and G are always complements to each other.

• Complete answer, your means which one is True / – user4249446 Apr 20 '15 at 19:08
• The second option most definitely ! – Plutonium smuggler Apr 20 '15 at 19:14
• is it possible make your answer a bit clearer? – user4249446 Apr 20 '15 at 19:16
• What part is not clear ? – Plutonium smuggler Apr 20 '15 at 19:17
• why III and IV is not true ? – user4249446 Apr 20 '15 at 19:27