Question
Draw the truth tables for AND, OR, and NOT gates. Also explain why NAND and NOR are called universal gates.
Solution — Step by Step
The NOT gate has one input and one output. It simply inverts the input signal.
| Input A | Output Y = A’ |
|---|---|
| 0 | 1 |
| 1 | 0 |
Output is 1 only when all inputs are 1. Think of it as: “A AND B must both agree.”
| Input A | Input B | Output Y = A·B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Output is 1 when any input is 1. Only fails (gives 0) when both inputs are 0.
| Input A | Input B | Output Y = A+B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
NAND = AND followed by NOT. The output is the complement of the AND output.
| Input A | Input B | Y = (A·B)‘ |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
NOR = OR followed by NOT. Output is 1 only when both inputs are 0.
| Input A | Input B | Y = (A+B)‘ |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
Why This Works
Every digital circuit in existence — phones, computers, calculators — is built from combinations of just these basic gates. The logic follows directly from Boolean algebra: AND is multiplication (0×1=0), OR is addition where 1+1=1, and NOT is complementation.
NAND and NOR are called universal gates because you can build ANY other gate using only NAND gates (or only NOR gates). This matters enormously in manufacturing — you only need to fabricate one type of transistor circuit, then combine copies of it to get whatever logic you need.
- NOT A = A NAND A (connect both inputs together)
- A AND B = (A NAND B) NAND (A NAND B)
- A OR B = (A NAND A) NAND (B NAND B)
This is why NAND and NOR appear repeatedly in CBSE Class 12 questions — the NCERT explicitly asks students to derive AND, OR, NOT from NAND alone.
Alternative Method — Verifying Universality of NOR
We can build the three basic gates from NOR alone, just like we did with NAND:
- NOT A = A NOR A
- A OR B = (A NOR B) NOR (A NOR B)
- A AND B = (A NOR A) NOR (B NOR B)
In board exams, “Show that NAND is a universal gate” typically means: draw the circuit diagrams for NOT, AND, and OR using only NAND gates with proper labelling. The truth table verification is a bonus that fetches full marks.
Common Mistake
Students confuse the NAND truth table with AND. They write NAND output as 0 for the first three rows and 1 for the last — exactly the AND table — forgetting the NOT step. NAND is the complement of AND. The only row where NAND gives 0 is when both inputs are 1 (i.e., A=1, B=1 → 0). Every other row gives 1.
A quick check: count the 1s. AND has one 1. NAND has three 1s. If your NAND table has only one 1, you’ve written AND by mistake.