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Logic

De Morgan's Laws

Lesson

De Morgan’s Laws tell you how to push a negation through an AND or OR. The connective flips.

The two laws

¬(p∧q)≡¬p∨¬q\neg(p \wedge q) \equiv \neg p \vee \neg q
¬(p∨q)≡¬p∧¬q\neg(p \vee q) \equiv \neg p \wedge \neg q

Rule of thumb: negate each part and flip the connective.

Worked example

“It’s NOT the case that (it’s sunny AND warm).” By De Morgan: “It’s not sunny OR it’s not warm.”

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

¬(T∧T)\neg(T \wedge T)

Problem 2

¬(T∧F)\neg(T \wedge F)

Problem 3

¬(F∨F)\neg(F \vee F)

Problem 4

¬(T∨F)\neg(T \vee F)

Practice

Standard problems matching the lesson.

Problem 5

¬T∨¬F (equivalent to ¬(T∧F))\neg T \vee \neg F \ \text{(equivalent to }\neg(T \wedge F)\text{)}

Problem 6

¬T∨¬T\neg T \vee \neg T

Problem 7

¬F∨¬F\neg F \vee \neg F

Problem 8

¬T∧¬F\neg T \wedge \neg F

Problem 9

¬F∧¬F\neg F \wedge \neg F

Problem 10

¬(F∧T)\neg(F \wedge T)

Problem 11

¬(F∧F)\neg(F \wedge F)

Problem 12

¬(T∨T)\neg(T \vee T)

Problem 13

¬(F∨T)\neg(F \vee T)

Problem 14

Apply DM: ¬(p∧q). Result is ¬p ? ¬q (1=∨, 0=∧)\text{Apply DM: } \neg(p \wedge q). \text{ Result is } \neg p \ ? \ \neg q \text{ (1=∨, 0=∧)}

Problem 15

Apply DM: ¬(p∨q). Connective (1=∨, 0=∧)\text{Apply DM: } \neg(p \vee q). \text{ Connective (1=∨, 0=∧)}

Problem 16

p=T,q=F; ¬(p∧q)p = T, q = F; \ \neg(p \wedge q)

Problem 17

p=T,q=F; ¬p∨¬qp = T, q = F; \ \neg p \vee \neg q

Problem 18

p=F,q=F; ¬(p∨q)p = F, q = F; \ \neg(p \vee q)

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

p=T,q=T,r=F; ¬(p∧q∧r)p = T, q = T, r = F; \ \neg(p \wedge q \wedge r)

Problem 20

p=T,q=T,r=T; ¬(p∧q∧r)p = T, q = T, r = T; \ \neg(p \wedge q \wedge r)

Problem 21

p=F,q=F,r=F; ¬(p∨q∨r)p = F, q = F, r = F; \ \neg(p \vee q \vee r)

Problem 22

p=T,q=F; ¬p∧¬qp = T, q = F; \ \neg p \wedge \neg q

Problem 23

p=T,q=T; ¬p∨¬qp = T, q = T; \ \neg p \vee \neg q

Problem 24

p=F,q=F; ¬p∧¬qp = F, q = F; \ \neg p \wedge \neg q

Problem 25

¬(¬p∨¬q) with p=T,q=T\neg(\neg p \vee \neg q) \text{ with } p=T, q=T

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