Practice Conditional Statement 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 conditional $P \to Q$ is a statement meaning "if $P$ is true, then $Q$ must be true," read as "if $P$ then $Q$ ."

A promise or rule: if the condition holds, the consequence follows.

Showing a random 20 of 50 problems.

Example 1

medium

Rewrite 'no odd number is divisible by 2' as a conditional.

Example 2

easy

Identify the hypothesis and conclusion in: 'If it rains, then the ground is wet.'

Example 3

hard

P \to Q

is true and

Q \to R

is true. By hypothetical syllogism, what conditional follows?

Example 4

hard

Determine the truth value: 'If

\sqrt{2}

is rational, then

\pi

is rational.'

Example 5

easy

For

P \to Q

with

P

false and

Q

true, what is its truth value?

Example 6

medium

Express 'being divisible by

4

is sufficient for being even' as a conditional.

Example 7

medium

If '

P \to Q

' is true and

P

is true, what can you conclude about

Q

Example 8

challenge

Find a quadrilateral that is a counterexample to the converse of: 'If a quadrilateral is a rhombus, then its diagonals are perpendicular.'

Example 9

medium

Identify the hypothesis and the conclusion in the statement: 'If a number is divisible by 4, then it is even.'

Example 10

medium

Express 'a positive number is necessary for the log to be defined' as a conditional.

Example 11

challenge

Prove that

P \to Q

is logically equivalent to

\neg P \lor Q

using truth values.

Example 12

easy

For

P \to Q

with

P

false and

Q

false, what is its truth value?

Example 13

easy

For

P \to Q

with

P

true and

Q

false, what is its truth value?

Example 14

easy

Rewrite 'every multiple of 4 is even' as an if-then conditional.

Example 15

challenge

Prove using a truth table that

P \to Q

and

\neg Q \to \neg P

are logically equivalent.

Example 16

medium

For statements

P, Q

, in how many of the four truth-value rows is

P \to Q

true?

Example 17

medium

Give a counterexample showing the converse of 'if

n

is a multiple of 4, then

n

is even' is false.

Example 18

easy

Write the inverse of

P \to Q

Example 19

medium

How many rows of a

2

-variable truth table make

P \to Q

false?

Example 20

hard

Affirming the consequent is invalid. Given

P \to Q

true and

Q

true, can we conclude

P

? Answer

1

for yes,

0

for no.

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Related Concepts

Logical Statement Contrapositive Biconditional