{"id":41220,"date":"2026-09-14T05:53:46","date_gmt":"2026-09-14T05:53:46","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=41220"},"modified":"2026-09-14T06:05:15","modified_gmt":"2026-09-14T06:05:15","slug":"de-morgans-law","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/de-morgans-law\/","title":{"rendered":"What is De Morgan’s Law? Definition, Rules & Examples"},"content":{"rendered":"\t\t
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\r\n Picture two bouncers standing at the door of a logical statement. One says \"NOT (this AND that).\" The other says \"NOT this, OR NOT that.\" They sound like they're arguing, but they're actually saying the exact same thing, just wearing different outfits. That's the whole trick behind DeMorgan's Law, and once you see it, you can't unsee it: it's hiding in your spam filter, your thermostat's circuit board, and the sentence you just used to talk your way out of dinner plans.\r\n\r\n\r\n\r\n<\/p>\r\n\r\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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What Is DeMorgan’s Law?<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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At its core, DeMorgan’s Law is a pair of rules in logic and set theory that describe how negation interacts with “and” (conjunction) and “or” (disjunction). Named after the 19th-century British mathematician Augustus De Morgan, the law shows that negating a compound statement flips both the connective and the components inside it.<\/span><\/p>

In plain English: “not (A and B)” is the same as “not A or not B.” And “not (A or B)” is the same as “not A and not B.” Negation doesn’t just cancel things out \u2014 it redistributes itself, like a courier who splits one package into two and flips a switch on each.<\/span><\/p>

Recommended Reading:<\/span> How to Write 99000 in Words? | Number in Words<\/a><\/strong><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\n De Morgan's Law in Boolean Algebra\r\n <\/h2>\r\n

\r\n De Morgan's Law explains how logical operations change when a statement is negated.\r\n It is widely used in Boolean algebra, digital logic, computer science, and programming.\r\n Let's explore its two simple and powerful rules.\r\n <\/p>\r\n <\/div>\r\n\r\n \r\n

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\r\n RULE 1\r\n <\/div>\r\n\r\n

\r\n Negation of a Conjunction\r\n <\/h3>\r\n\r\n
\r\n \u00ac(A \u2227 B) = (\u00acA) \u2228 (\u00acB)\r\n <\/div>\r\n\r\n

\r\n In simple words: NOT (A AND B)<\/strong> becomes\r\n (NOT A) OR (NOT B)<\/strong>.\r\n <\/p>\r\n <\/div>\r\n\r\n \r\n

\r\n
\r\n RULE 2\r\n <\/div>\r\n\r\n

\r\n Negation of a Disjunction\r\n <\/h3>\r\n\r\n
\r\n \u00ac(A \u2228 B) = (\u00acA) \u2227 (\u00acB)\r\n <\/div>\r\n\r\n

\r\n In simple words: NOT (A OR B)<\/strong> becomes\r\n (NOT A) AND (NOT B)<\/strong>.\r\n <\/p>\r\n <\/div>\r\n\r\n \r\n

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\r\n \u2726 De Morgan's Laws in Boolean Notation\r\n <\/h3>\r\n\r\n
\r\n\r\n
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\r\n AND \u2192 OR\r\n <\/div>\r\n
\r\n (A \u00b7 B)' = A' + B'\r\n <\/div>\r\n <\/div>\r\n\r\n
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\r\n OR \u2192 AND\r\n <\/div>\r\n
\r\n (A + B)' = A' \u00b7 B'\r\n <\/div>\r\n <\/div>\r\n\r\n <\/div>\r\n <\/div>\r\n\r\n \r\n
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\r\n \ud83d\udca1 Easy Way to Remember\r\n <\/h3>\r\n

\r\n When the whole expression is negated, AND changes to OR<\/strong>,\r\n while OR changes to AND<\/strong>. Each individual statement is also negated.\r\n <\/p>\r\n <\/div>\r\n\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\n De Morgan's Law Proof: Showing the Symmetry Holds\r\n <\/h2>\r\n\r\n

\r\n De Morgan's Law can be verified using a truth table. By checking every\r\n possible combination of A and B, we can see that both sides of the\r\n equation always produce the same result.\r\n <\/p>\r\n\r\n <\/div>\r\n\r\n \r\n

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\r\n Proof by Truth Table\r\n <\/h3>\r\n\r\n

\r\n Let's verify the first De Morgan's Law:\r\n <\/p>\r\n\r\n

\r\n \u00ac(A \u2228 B) = \u00acA \u2227 \u00acB\r\n <\/div>\r\n\r\n <\/div>\r\n\r\n \r\n
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A<\/th>\r\n B<\/th>\r\n \u00acA<\/th>\r\n \u00acB<\/th>\r\n A \u2228 B<\/th>\r\n \u00ac(A \u2228 B)<\/th>\r\n \u00acA \u2227 \u00acB<\/th>\r\n <\/tr>\r\n <\/thead>\r\n\r\n
1<\/td>\r\n 1<\/td>\r\n 0<\/td>\r\n 0<\/td>\r\n 1<\/td>\r\n 0<\/td>\r\n 0<\/td>\r\n <\/tr>\r\n\r\n
1<\/td>\r\n 0<\/td>\r\n 0<\/td>\r\n 1<\/td>\r\n 1<\/td>\r\n 0<\/td>\r\n 0<\/td>\r\n <\/tr>\r\n\r\n
0<\/td>\r\n 1<\/td>\r\n 1<\/td>\r\n 0<\/td>\r\n 1<\/td>\r\n 0<\/td>\r\n 0<\/td>\r\n <\/tr>\r\n\r\n
0<\/td>\r\n 0<\/td>\r\n 1<\/td>\r\n 1<\/td>\r\n 0<\/td>\r\n 1<\/td>\r\n 1<\/td>\r\n <\/tr>\r\n <\/tbody>\r\n\r\n <\/table>\r\n <\/div>\r\n\r\n \r\n
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\r\n \u2713 What Does the Table Show?\r\n <\/h3>\r\n\r\n

\r\n The final two columns are identical in every row. This confirms that\r\n \u00ac(A \u2228 B) = \u00acA \u2227 \u00acB<\/strong> is true for every possible combination\r\n of A and B.\r\n <\/p>\r\n <\/div>\r\n\r\n \r\n

\r\n

\r\n Try the Second Identity\r\n <\/h3>\r\n\r\n
\r\n \u00ac(A \u2227 B) = \u00acA \u2228 \u00acB\r\n <\/div>\r\n\r\n

\r\n Create the same truth table for this expression and compare the final\r\n columns. They will match perfectly!\r\n <\/p>\r\n <\/div>\r\n\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Why It’s Named After a Man Who Didn’t Even Discover It First<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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“Every act of thought is an act of measurement.” \u2014 Augustus De Morgan<\/span><\/i><\/span><\/p>

Here’s a delicious irony: versions of these rules trace back to ancient Greek and medieval logicians centuries before Augustus De Morgan was born. But De Morgan was the one who formalized them rigorously within symbolic logic in the mid-1800s, and mathematics, much like a stubborn group chat,\u00a0 tends to name things after whoever shows up with the receipts, not whoever had the idea first.<\/span><\/p>

Recommended Reading:<\/span> What Are the Measures of Central Tendency in Math?<\/a><\/strong><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Where DeMorgan’s Law Shows Up in the Real World<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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This isn’t just chalkboard trivia. DeMorgan’s Law is quietly load-bearing infrastructure across several fields:<\/span><\/p>