Warm-up: What is the definition of a “birthday?” Can you write the “definition” or rules of declaring a day someone’s birthday as an “If… Then…” statement (conditional)?
Hopefully your answers look something like this:
A birthday is the day that someone was born.
If some one was born on a certain day, then that day is his/her birthday.
If a day is someone’s birthday, then he/she was born on that day.
Either of the above conditionals are valid ways to describe the definition of a “birthday.” Take a moment to think about the relationship between those two statements though. Although they say pretty much the same thing, they are actually converses of one another (switched the hypothesis and conclusion).
Hopefully you realize that it is a special case for a statement and its converse to have the same truth value. If you think back to “If it is raining then there are clouds above,” its converse does not have the same truth value as the original statement (I’ll leave that for you to verify).
Special conditional statements, where both the original conditional and its converse have the same truth value, are called biconditional statements.
p <–> q which is read “p if and only if q”,
p <–> q means p –> q and q –> p.
Instead of writing down the whole phrase “if and only if,” lazy mathematicians agreed to just write “iff,” so when you’re reading a text and you come across “iff” just read it as “if and only if“
Mathematical language: Conditional statement is true and its converse is true.
Everyday terms: think “vice versa”- If today is my birthday, then I was born today, and “vice versa.” If you can say “vice versa” at the end of a statement, then it’s probably a biconditional statement.
This first example needs some set up. When I was in High School, every Friday was spicy chicken patty day. I wasn’t crazy about it, but some people were basically religious about spicy chicken patty day. We also need to realize, that you can’t get my high school’s chicken patty just anytime or anywhere. Only Friday’s, only in the cafeteria. I would feel comfortable saying, about my friend Pat for example, that “If it was Friday, then Pat was eating a spicy chicken patty for lunch.” I could also say, conversely, “If Pat is eating a spicy chicken patty for lunch, then it is Friday.” Since the original conditional and its converse are both true, we know that we have a biconditional statement. Let’s rewrite it in biconditional form though:
Pat eats spicy chicken patties if and only if it is Friday.
Remember this: biconditional statements use the phrase if and only if or the abbreviation iff
Next we talk about the types of problems or situations that you will encounter with biconditional statements.
Rewrite a definition in two converse, conditional forms and in biconditional form:
Ex 1.) Obtuse angle- angle whose measure is greater than 90º and less than 180º.
Given a statement, look at its converse, rewrite as a biconditional if there is one, provide a counterexample if the statement is not biconditional.
Ex 2.)If a rectangle’s sides are all the same length, then the rectangle is a square.
Ex 3.) If a two numbers are positive, then their product is positive.
Given a potential biconditional, check the truth-value of the conditional statement and its converse. Provide a counterexample if the truth-value is false.
Ex 4.) A rectangle has an area of 24m2 iff it has sides of 6m and 4m.
Ex 5.) A square has an area of 25m2 iff it has sides of 5m2.
Wrap-up- Hopefully, without looking back, you can answer these questions:
How do we check whether or not a statement is biconditional?
Definitions can usually be rewritten as what kind of statements?
What is the key “phrase” of a biconditional statement?
What does the Venn diagram of a biconditional look like?
Now use your textbook to get some further practice.