Inverses, converses, and contrapositives

Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

Get the most by viewing this topic in your current grade. Pick your course now.

  1. The inverse, converse, and contrapositive Overview:
  2. Inverse Statements
  3. Converse Statements
  4. Contrapositive Statements
  5. Logical equivalents
  1. Finding the inverse, converse and contrapositive
    Given the statements, write the inverse, converse, and contrapositive:
    1. Two intersecting lines create an angle.
    2. If today is Monday, then Kevin will play soccer.
    3. If 1+2=31+2=3, then 12+22=321^2+2^2=3^2.
    4. If the polygon is a triangle, then it has 3 sides.
  2. For each statement, write the inverse, converse, and contrapositive in symbolic form:
    1. rsr \to s
    2. pp toto ~qq
    3. ~mm \to nn
    4. ~uu \to ~vv
  3. Truth value of inverse, converse and contrapositive
    Write the converse, and find the truth value of the converse:
    1. If x+7=13x+7=13, then x=6x=6.
    2. If 33 is odd, then 3+13+1 is even.
    3. If 22 is an integer, then 22 is a whole number.
  4. Assume that the conditional statement is true. Write the inverse and state whether the inverse is always true, sometimes true, or never true:
    1. pqp \to q
    2. q topq\ to p
    3. ~qq \to ~pp
    4. pp \to ~qq
  5. Logical Equivalents

    Find the truth value of the following conditionals. Then write the contrapositive and find the truth value of the contrapositive. Are the truth values the same?

    1. If (2+3)×4=20, then 2+3=52+3=5.
    2. If 44 is even, then 4+24+2 is even.
    3. If 33 is an integer, then 33 is not a whole number.
  6. Finding truth values of original statements

    Find possible truth values for pp and qq where:

    1. pqp \to q and ~qq \to ~pp is both false?
    2. pp \to qq and qq \to pp is both true?
    3. pqp \to q and qpq \to p is both false?
    4. qpq \to p and ~qq \to ~pp is both true?
Free to Join!
StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.
  • Easily See Your Progress

    We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
  • Make Use of Our Learning Aids

    Last Viewed
    Practice Accuracy
    Suggested Tasks

    Get quick access to the topic you're currently learning.

    See how well your practice sessions are going over time.

    Stay on track with our daily recommendations.

  • Earn Achievements as You Learn

    Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
  • Create and Customize Your Avatar

    Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
Topic Notes

Let pp be the hypothesis and qq be the conclusion. Then:

An inverse statement is formed by negating both the hypothesis and conclusion of the conditional. In symbolic form it would be:

~pp \to ~qq

A converse statement is formed by switching the hypothesis and the conclusion of the conditional. In symbolic form it would be:

q pq\ \to p

A contrapositive statement is formed by negating both the hypothesis and conclusion, AND switching them. In symbolic form it would be:

~qq \to ~pp

Statements which always have the same truth values are logical equivalents.
Conditionals and contrapositives are logical equivalents, and inverses and converses are logical equivalents.