Practice Biconditional in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

medium

Write the negation of

P\leftrightarrow Q

in terms of XOR.

Example 3

hard

Is the biconditional commutative? Justify.

Example 4

medium

Is the statement '

x>0

iff

|x|=x

' true for all real

x

Example 5

easy

Read aloud the symbol

\leftrightarrow

in two common ways.

Example 6

easy

Does '

P

Q

' mean the same as '

P

if and only if

Q

Example 7

hard

For real

x

, is '

x^2\ge 0

iff

x

is real' a true statement?

Example 8

easy

Both

P

and

Q

are true. Is

P \leftrightarrow Q

true or false?

Example 9

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 10

medium

Verify that

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

using a truth table.

Example 11

easy

Evaluate the biconditional

p \Leftrightarrow q

for all truth value combinations and construct its truth table.

Example 12

medium

A square is a rectangle iff ____.

Example 13

medium

How many distinct true rows does the truth table of

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

have for three variables?

Example 14

hard

Is the statement 'a triangle is right-angled iff

a^2+b^2=c^2

for some labeling of sides' true?

Example 15

medium

True or false: in classical logic,

P\leftrightarrow Q

is logically equivalent to

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

Example 16

medium

Simplify

(P \leftrightarrow F)

where

F

is a contradiction.

Example 17

hard

How many of the 16 binary connectives on

\{P,Q\}

are equivalent to

P\leftrightarrow Q

Example 18

medium

Given

P \leftrightarrow Q

is true and

Q

is true, what is the truth value of

P

Example 19

medium

Determine whether '

n

is even

\Leftrightarrow

n^2

is even' is true for all integers

n

Example 20

easy

In the truth table of

P \leftrightarrow Q

, how many of the four rows are true?

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Related Concepts

Conditional Statement