Usually in life, when you think about a certain event, there were some other events surrounding it that allowed it to happen. You could call the things necessary for an event to occur the conditions under which the event can occur.
Ex 1: Think about the conditions under which you would go camping.
Conditional statements are often represented using an “If…then…” format, and include two parts called the hypothesis and conclusion.
It is important to note that conditional statements don’t imply cause and effect.
Ex 2: think about the conditional statement “If it is raining, then there are clouds above.”
Here, the hypothesis, or condition, being talked about is that it is raining.
According to the conditional statement, if it is raining, then we are safe to come to the conclusion that there also must be clouds above.
Hypothesis: It is raining.
Conclusion: There are clouds overhead.
Sometimes you will be asked to take information and rewrite it as a conditional statement in “If…then…” form.
Ex 3: It only rains when it is cloudy.
We can rewrite this as the conditional statement: “If it is raining, then it is cloudy”
Ex 4: Conditional statements can also be represented using a Venn diagram.
From this Venn diagram, we can see that there are lots of cases where it is cloudy, but not raining, but all of the cases where it is raining occur only when it is cloudy.
Converse, Inverse, and Contrapositive
From a given conditional statement, it is useful to consider 3 other statements related to it.
Think about how the following statements are similar and/or different to the conditional statement “If it is raining, then there are clouds overhead.”
1. If there are clouds overhead, then it is raining.
2. If it is not raining, then there are not clouds overhead.
3. If there are not clouds overhead, then it is not raining.
How was the original conditional statement changed to get statements 1, 2, and 3?
Statement 1 switched the hypothesis and conclusion of the original conditional statement. Statement 1 is the converse statement of the original conditional statement.
Statement 2 negated both the hypothesis and conclusion. Statement 2 is the inverse statement of the original conditional statement.
Statement 3 both switched and negated the hypothesis and conclusion. Statement 3 is the contrapositive statement of the original conditional statement.