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Conditional statements are all part of logic. Conditional statements are also called 'if-then' statements. A conditional statement can be written in the form, "If p, then q." So, logically, it makes sense that they are called 'if-then' statements!

p is the hypothesis of the statement
q is the conclusion of the statement

Sometimes a graphic can make things clearer.

conditionalstatements

If a conditional statement is true, the conclusion is always true when the hypothesis is true.

Can you see why this is the case? The Venn Diagram shows the relationship between p and q

The hypothesis is the part that follows the p or the if
The conclusion is the part that follows the q or the then

We can take a look at it with some statements in the Venn and that may make it even clearer!

Let's take a look at this example…

conditionalstatements1

Here's our conditional statement, "If a fruit is a banana, then it grows on a tree."

hypothesis (p): bananas are a fruit
conclusion (q): they grow on a tree

The Venn Diagram shows clearly that all bananas are fruits that grow on trees.

In math 'terms,' you can write this as

conditionalstatementsedited1

and, this is read, "If p then q."

Converse

Now, you may be wondering how to write the converse of the conditional statement, "If a fruit is a banana, then it grows on a tree." It is actually quite simple! The converse is simply taking the q and reversing it with the p! In other words,

conditionalstatementsedited1a

and, this is read, "If q then p."

In this case, the converse of the statement is, "If it grows on a tree, then the fruit is a banana." Get it? Just flip the conclusion and the hypothesis around! It's that easy!

What if the conditional statement was, "If a dog is a Dachshund, then it is a wiener dog." The converse would be, "If a dog is a wiener dog, then it is a Dachshund."

Inverse

How about the inverse of the conditional statement, "If a fruit is a banana, then it grows on a tree." Think about what you already know about inverses…what does inverse mean? It means the OPPOSITE, right? So, take the opposite of the p and the opposite of the q and voila! You've got the inverse!

conditionalstatementsedited1b

This is read, "If not p then not q."

So the inverse of the conditional statement, "If a fruit is a banana, then it grows on a tree" becomes "If a fruit is NOT a banana, then it does NOT grow on a tree."

The wiener dog conditional statement, "If a dog is a Dachshund, then it is a wiener dog" becomes "If a dog is not a Dachshund, then it is not a wiener dog."

Contrapositive

How do you write the contrapositive of a conditional statement? So, going with our statement "If a fruit is a banana, then it grows on a tree." The contrapositive does the job of both the inverse AND the converse. In other words…

conditionalstatementsedited1c

This is read, "If not q then not p."

Can you figure out what the contrapositive of our banana statement would be? Did you say, "If it doesn't grow on a tree, then it isn't a banana?" If you did, well then, you are correct! Yahooo!

©2016 Sherry Skipper Spurgeon for Geometry Bugs Me

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Conditional Statements