Biconditional Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Biconditional.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A biconditional $P \leftrightarrow Q$ is true when $P$ and $Q$ have the same truth value — both true or both false.

' $P$ if and only if $Q$ '—they're equivalent, true together or false together.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A biconditional P iff Q is true exactly when P and Q share the same truth value.

Common stuck point: The procedure for biconditional is the easy part; the trap is proving one direction and calling it 'iff'. Asking "Are both statements true together and false together, in every case?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are both statements true together and false together, in every case?

Worked Examples

Example 1

easy

Evaluate the biconditional

p \Leftrightarrow q

for all truth value combinations and construct its truth table.

Answer

\begin{array}{cc|c}p & q & p \Leftrightarrow q\\ \hline T&T&T\\T&F&F\\F&T&F\\F&F&T\end{array}

First step

p \Leftrightarrow q

means '

p

if and only if

q

' — it is true when

p

and

q

have the same truth value.

Full solution

2
Row $(T,T)$ : both true — same value — $T$ .
3
Row $(T,F)$ : different values — $F$ .
4
Row $(F,T)$ : different values — $F$ .
5
Row $(F,F)$ : both false — same value — $T$ .

A biconditional is true precisely when both sides share the same truth value. It is equivalent to

(p \Rightarrow q) \land (q \Rightarrow p)

Example 2

medium

Determine whether '

n

is even

\Leftrightarrow

n^2

is even' is true for all integers

n

Example 3

medium

Construct a truth table for

(P \leftrightarrow Q) \land P

Example 4

hard

How many of the 16 binary connectives on

\{P,Q\}

are equivalent to

P\leftrightarrow Q

Example 5

challenge

Prove: for integers

a,b

, '

a\equiv b \pmod 5

iff

5 \mid (a-b)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

State whether each biconditional is true or false: (a) '

3 = 3 \Leftrightarrow 5 > 2

', (b) '

3 = 4 \Leftrightarrow 1 = 2

', (c) '

3 = 3 \Leftrightarrow 1 = 2

Example 2

medium

Verify that

p \Leftrightarrow q \equiv (p \Rightarrow q) \land (q \Rightarrow p)

using a truth table.

Example 3

easy

Is the biconditional '

2+2=4

if and only if

3

is odd' true or false?

Example 4

easy

Is '

5>3

if and only if

1=2

' true or false?

Example 5

easy

Both

P

and

Q

are false. Is

P \leftrightarrow Q

true or false?

Example 6

easy

Translate '

P

if and only if

Q

' into implication arrows.

Example 7

easy

'A triangle is equilateral if and only if all three angles equal

60^\circ

.' How many directions must a proof establish?

Example 8

easy

Does '

P

Q

' mean the same as '

P

if and only if

Q

Example 9

easy

Both

P

and

Q

are true. Is

P \leftrightarrow Q

true or false?

Example 10

easy

In the truth table of

P \leftrightarrow Q

, how many of the four rows are true?

Example 11

medium

For which truth values of

P

and

Q

P \leftrightarrow Q

FALSE? List all.

Example 12

medium

Is the statement '

x^2=4

if and only if

x=2

' true for all real

x

? Why?

Example 13

medium

Rewrite the definition 'an integer is even iff it is divisible by

2

' as two conditional statements.

Example 14

medium

P \leftrightarrow Q

is true and

P

is false, what is the truth value of

Q

Example 15

medium

P \leftrightarrow Q

logically equivalent to

\lnot P \leftrightarrow \lnot Q

Example 16

medium

Express the biconditional

P \leftrightarrow Q

using only XOR (

\oplus

) and negation.

Example 17

medium

A student proves '

n

even

\Rightarrow n^2

even' and claims they proved '

n

even iff

n^2

even.' What is missing?

Example 18

medium

Determine the truth value of

(P \leftrightarrow Q) \leftrightarrow (Q \leftrightarrow P)

for all

P,Q

Example 19

medium

Given

P \leftrightarrow Q

is true and

Q

is true, what is the truth value of

P

Example 20

challenge

Show that

P \leftrightarrow Q

is equivalent to

(P \land Q) \lor (\lnot P \land \lnot Q)

, then state the truth value when

P

is true and

Q

is false.

Example 21

challenge

Prove or disprove:

((P \leftrightarrow Q) \leftrightarrow R)

is equivalent to

(P \leftrightarrow (Q \leftrightarrow R))

Example 22

challenge

A definition states '

x

is a maximum of set

S

iff

x \in S

and

x \ge s

for all

s \in S

.' Using the biconditional, what TWO facts can you derive once you know

x

is the maximum?

Example 23

easy

Evaluate '

\pi>3

iff

7

is prime.'

Example 24

easy

Evaluate '

2+2=5

iff the moon is made of cheese.'

Example 25

easy

Evaluate '

1+1=2

iff

0>1

Example 26

medium

Is the statement '

x>0

iff

|x|=x

' true for all real

x

Example 27

medium

Rewrite '

n

is divisible by 3 iff the sum of digits of

n

is divisible by 3' as two implications.

Example 28

medium

(P \to Q) \leftrightarrow (\lnot Q \to \lnot P)

a tautology?

Example 29

medium

Simplify

(P \leftrightarrow T)

where

T

is a tautology.

Example 30

medium

Simplify

(P \leftrightarrow F)

where

F

is a contradiction.

Example 31

medium

How many distinct true rows does the truth table of

(P \leftrightarrow Q) \lor (P \leftrightarrow R)

have for three variables?

Example 32

medium

Write the negation of

P\leftrightarrow Q

in terms of XOR.

Example 33

medium

A square is a rectangle iff ____.

Example 34

medium

Translate to symbols: 'You pass iff you score at least 60.' Let

P

= pass,

S

= score

\ge 60

Example 35

hard

Show that

(P\leftrightarrow Q)\land(Q\leftrightarrow R)

implies

P\leftrightarrow R

Example 36

hard

Is the biconditional commutative? Justify.

Example 37

hard

For real

x

, is '

x^2\ge 0

iff

x

is real' a true statement?

Example 38

hard

Is the statement 'a triangle is right-angled iff

a^2+b^2=c^2

for some labeling of sides' true?

Example 39

hard

For sets

A,B

A=B

iff ____ (state the standard set-equality biconditional).

Example 40

challenge

Determine whether

P\leftrightarrow(Q\leftrightarrow R)

equals

(P\leftrightarrow Q)\leftrightarrow R

by counting true rows.

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Related Concepts

Conditional Statement

Background Knowledge

These ideas may be useful before you work through the harder examples.

conditional