Why is $ab + bc + c\bar{a} = ab + c\bar{a}$ true in binary?

Question

I was simplifying the equation of a logic gates problem and I realized that ab + bc + cā and ab + cā followed the same truth table which is the following:

-----------------
| A | B | C | S |
-----------------
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
-----------------

The problem is that I haven't found any way to get to the simplified expression from the main one by applying boolean algebra procedures. Any idea? Thanks.

Answer 1

Another way to illustrate this equivalence is a Karnaugh-Veitch map:

The "green" term $b c$ can be eliminated, because it is covered by the disjunction of $c a'$ and $a b$.