Bubble pushing

Also called: alternative gate symbols, alternate gate symbols, De Morgan equivalent symbols, De Morgan equivalent gates, dual gate symbols

Redrawing a gate or circuit by swapping AND and OR bodies and toggling every bubble. The function stays the same; it is De Morgan's law in pictures.

Bubble pushing is a way to redraw a gate in a second, equivalent form, or to simplify a circuit full of inversions, without writing any algebra.

The recipe for one gate has two steps, and you need both:

Swap the body: AND becomes OR, OR becomes AND.
Toggle every bubble: on each input and on the output, add a bubble where there wasn't one and remove it where there was.

It works because it is De Morgan's law drawn as a picture. = says a NAND (AND body, output bubble) equals an OR body with bubbled inputs.

The four pairs worth knowing:

NAND = OR with bubbled inputs.
NOR = AND with bubbled inputs.
AND = OR with bubbles on the inputs and the output.
OR = AND with bubbles on the inputs and the output.

The second tool is double inversion: two bubbles on the same wire cancel. So you may add a pair of bubbles to any wire, or remove a pair, freely.

Put together, these turn an AND-OR circuit into nand nand logic and an OR-AND circuit into nor nor logic. They also make diagrams easier to read: a NAND drawn as a bubbled OR says plainly "1 if any input is 0".

Worked examples

Example

Redrawing a NOR

Bubble-push a 2-input NOR and check the result. The circuit above draws the answer with explicit inverters.


0	0	1	1
0	1	0	0
1	0	0	0
1	1	0	0

1.
Start: a NOR is an OR body, no input bubbles, one output bubble.
2.
Swap the body: OR becomes AND.
3.
Toggle every bubble: the output bubble goes, and each input gains one.
4.
Result: an AND with bubbled inputs, which computes .
5.
Check all rows 00 to 11. NOR gives 1 0 0 0. gives 1·1 = 1, 1·0 = 0, 0·1 = 0, 0·0 = 0: also 1 0 0 0. ✓

Example

Spotting half a push

A student redraws a NOR as an OR with bubbled inputs, keeping the OR body. Test it.

1.
An OR with bubbled inputs computes .
2.
Try A = 1, B = 0: = 0 + 1 = 1, but NOR(1, 0) = 0. They disagree.
3.
The student moved the bubbles but skipped step 1. is actually NAND.
4.
Doing both steps gives an AND body with bubbled inputs, , which matches NOR in every row.

Common mistakes

Moving the bubbles but keeping the body. An AND with bubbled inputs is a NOR, not a NAND.
Swapping the body but forgetting to toggle one of the bubbles, usually the output one.
Cancelling bubbles that sit on different wires. Only two bubbles on the same wire cancel.

Practice Bubble pushing

Interactive questions with instant feedback and a worked solution for every wrong answer.

Logic Gates lesson

An AND gate is drawn with a bubble on each input and a bubble on its output. Which single standard gate is it equivalent to?

Practice exams

Learn it step by step

Bubble pushing is taught in Logic Gates.

Logic Gates →

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