Combine terms for single rows into larger terms in truth table

Question

I'm wondering what does it mean by combining terms for single rows into larger terms in a truth table. Let's say: (ABC are inputs and F is output)

A | B | C | F
0   0   0   0
0   0   1   0
0   1   0   1
0   1   1   1
1   0   0   0
1   0   1   1
1   1   0   0
1   1   1   1

Then:

F = A'BC'+A'BC+AB'C+ABC
  = A'B+AC

Is this how I combine the terms for single rows into larger terms?

Answer 1

The relevant equivalence is called:

Adjacency

$PQ + PQ' = P$

(And its dual) $(P+Q)(P+Q')=P$

It's called adjacency as in a Karnaugh diagram (or K-map) the squares you combine into larger blocks are (typically) adjacent to each other. If you have not seen K maps yet, I encourage you to take a look at those and much will become clear! Note that in a truth-table the lines that are being combined are somtime next to its other as well (as with your rows 3 and 4) but don't have to be (as with your rows 6 and 8), but in both cases the only difference between the two rows being combined is the value of one variable (the $Q$ vs $Q'$ in the Adjancency Law as stated). K maps are a way way to re-express the tuth-conditions of a truth-function so that any such combinations can more easily be found' and thus boolean expressions more easily be simplified.

Finally, Adjacency can be reduced to (proven from) more basic principles:

$PQ + PQ' =$ (Distribution)

$P(Q +Q')=$ (Complement)

$P1 = $ (Identity)

$P$