Question

For Boolean variables A, B and C, list all the possible products. (Note that order of literals doesn’t matter. For example, B A is considered to be the same product as A B, and the same is true for ABC and BCA.)

Answer 1

We have three variables A,B,C so the possible products can be : ABC,ACB,BAC,BCA,CAB,CBA

Using the truth table method we can show:

A	B	C	AB	ABC	AC	ACB	BA	BAC	BC	BCA	CA	CAB	CB	CBA
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
0	1	1	0	0	0	0	0	0	1	0	0	0	1	0
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	0	0	1	0	0	0	0	0	1	0	0	0
1	1	0	1	0	0	0	1	0	0	0	0	0	0	0
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1

For the two variable product pairs we can clearly see from the truth table that:

AB = BA, AC = CA, BC = CB

Similarly for three variable products :

ABC = ACB = BAC = BCA = CAB = CBA

Hence proved that ABC = BCA