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Logic

Conditional Statements

Lesson

A conditional p→qp \to q (“if p then q”) is false only when p is true and q is false. The promise is broken only by a true premise leading to a false conclusion.

Truth table

  p | q | p→q
 ---+---+----
  T | T |  T
  T | F |  F
  F | T |  T
  F | F |  T

Vacuous truth

When pp is false, the conditional is trueby default. “If pigs fly, then 2+2=5” is true because pigs don’t fly.

Worked example

“If it’s raining, then the ground is wet.”

Raining and ground wet → T. Raining but ground dry → F. Not raining, ground either way → T (no broken promise).

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

T→TT \to T

Problem 2

T→FT \to F

Problem 3

F→TF \to T

Problem 4

F→FF \to F

Practice

Standard problems matching the lesson.

Problem 5

“If 2+2=4, then 3<5”\text{``If 2+2=4, then 3<5''}

Problem 6

“If 2+2=4, then 3>5”\text{``If 2+2=4, then 3>5''}

Problem 7

“If 2+2=5, then 3<5”\text{``If 2+2=5, then 3<5''}

Problem 8

“If 2+2=5, then 3>5”\text{``If 2+2=5, then 3>5''}

Problem 9

For p→q to be F, p must be?\text{For } p \to q \text{ to be F, } p \text{ must be?}

Problem 10

For p→q to be F, q must be?\text{For } p \to q \text{ to be F, } q \text{ must be?}

Problem 11

T→¬TT \to \neg T

Problem 12

F→¬TF \to \neg T

Problem 13

¬F→T\neg F \to T

Problem 14

¬T→F\neg T \to F

Problem 15

T→(F∧T)T \to (F \wedge T)

Problem 16

T→(F∨T)T \to (F \vee T)

Problem 17

“If pigs fly, then 2+2=5” (pigs don’t fly)\text{``If pigs fly, then 2+2=5'' (pigs don't fly)}

Problem 18

“If 7 is prime, then 7 is even”\text{``If 7 is prime, then 7 is even''}

Challenge

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

Problem 19

(T∧T)→F(T \wedge T) \to F

Problem 20

(T∧F)→F(T \wedge F) \to F

Problem 21

(F∨T)→T(F \vee T) \to T

Problem 22

¬(T→F)\neg(T \to F)

Problem 23

¬(F→T)\neg(F \to T)

Problem 24

(T→T)→F(T \to T) \to F

Problem 25

(F→F)→T(F \to F) \to T

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