I have an overload of questions on digital logic. Hope that is okay!
1. Truth-functional completeness often makes circuits more complicated than they have to be. Is there anything besides cost-effectiveness that is beneficial about truth-functional completeness?
I assume you are asking “Why do we learn how to use NOR gates or NAND gates exclusively in a circuit?” Primarily, it is just to teach students how the gates work. But in practice, if you are constructing an electronic device, you may not have all the gates available (e.g. Radio Shack ran out and will not get them in for ten days), and so need to use a couple of NAND gates to do the job of one AND gate. Also, you might only use one NOR gate in the circuit, but a chip might contain four NOR gates, so why not just use three of them to replace an AND gate instead of buying one?
2. Since the symbol for NOR is the upside-down triangle, is there a symbol for NAND?
The triangle symbol is largely my own convention. See the Wikipedia page for the standard ways of expressing NOR. I know of no special symbol for NAND.
3. Is there a conditional gate (P ⊃ Q)?
Not that I am aware of. You can make a conditional using other gates.
4. Why do we write the names of logic gates in all caps? ( AND instead of and or And)
Just to distinguish them from the words in a sentence. It would be confusing to read “Take an or gate and an and gate…”
5. Why in K-maps do we circle in groups in powers of two?
Because that’s how they work to correctly simplify propositions. Draw yourself a K-map with 0111 across the top four cells, and 1110 across the bottom four cells. If you made two circles with groups of three across and ask, “What variable stays the same (negated or unnegated)?” the answer is that nothing stays the same, so no proposition can be identified. To get the simplest proposition, you must circle the middle four square, and the two on the top right and bottom left. Spend some time thinking through exactly what the K-map is doing when you circle groups and determine the proposition from the circled group. (See the next question.)
6. Finally, do K-maps eliminate the need for the Algebraic identities? I found that doing the Digital Logic Project didn’t require using them.
Yes, that is their primary benefit. Consider the proposition (p • q) ∨ (p • ~q). This simplifies this way:
- (p • q) ∨ (p • ~q) Given
- p • (q ∨ ~q) Distribution
- p • 1 Tautology
- p Alg. identity
Now do a 2×2 K-map for this proposition:
See how it does the same thing in a faster, easier way?