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Logic

Biconditional (If and Only If)

Lesson

A biconditional p↔qp \leftrightarrow q(“p if and only if q”) is true when both parts have the same truth value — both T or both F.

Truth table

  p | q | p↔q
 ---+---+----
  T | T |  T
  T | F |  F
  F | T |  F
  F | F |  T

Equivalent to (p→q)∧(q→p)(p \to q) \wedge (q \to p): two conditionals, one in each direction.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

T↔TT \leftrightarrow T

Problem 2

T↔FT \leftrightarrow F

Problem 3

F↔TF \leftrightarrow T

Problem 4

F↔FF \leftrightarrow F

Practice

Standard problems matching the lesson.

Problem 5

For p↔q to be T, parts must be...?\text{For } p \leftrightarrow q \text{ to be T, parts must be...?}

Problem 6

¬T↔F\neg T \leftrightarrow F

Problem 7

¬F↔T\neg F \leftrightarrow T

Problem 8

T↔¬TT \leftrightarrow \neg T

Problem 9

F↔¬FF \leftrightarrow \neg F

Problem 10

(T∧F)↔F(T \wedge F) \leftrightarrow F

Problem 11

(T∨F)↔T(T \vee F) \leftrightarrow T

Problem 12

(T→F)↔F(T \to F) \leftrightarrow F

Problem 13

(F→T)↔T(F \to T) \leftrightarrow T

Problem 14

T↔T↔TT \leftrightarrow T \leftrightarrow T

Problem 15

T↔T↔FT \leftrightarrow T \leftrightarrow F

Problem 16

“2+2=4 iff 3<5”\text{``2+2=4 iff 3<5''}

Problem 17

“2+2=4 iff 3>5”\text{``2+2=4 iff 3>5''}

Problem 18

“2+2=5 iff 3>5”\text{``2+2=5 iff 3>5''}

Challenge

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

Problem 19

¬(T↔F)\neg(T \leftrightarrow F)

Problem 20

¬(T↔T)\neg(T \leftrightarrow T)

Problem 21

(T↔F)↔F(T \leftrightarrow F) \leftrightarrow F

Problem 22

(T↔T)↔T(T \leftrightarrow T) \leftrightarrow T

Problem 23

(F↔F)↔F(F \leftrightarrow F) \leftrightarrow F

Problem 24

(T→T)↔(F→F)(T \to T) \leftrightarrow (F \to F)

Problem 25

(T→F)↔(F→T)(T \to F) \leftrightarrow (F \to T)

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