If-then statements are very common outside formal mathematics. In many cases, your familiarity with these types of statements will help you to interpret their meaning and truth value. In other cases, the truth value of some statements may be unclear without a formal understanding of logic. Consider the following two statements:
- If it rains today then you will stay at home and read a book.
- You stayed at home and read a book.
Did it or did it not rain?
Watch This
http://www.youtube.com/watch?v=oEr27P1bX9o James Sousa: If-Then Statements and Converses
Guidance
A mathematical set is just a group of things. The group can include anything: letters, numbers, objects or monkeys. Set theory focuses on the relationships between sets as they overlap or are completely within each other. For the purposes of if-then statements, set theory provides a perfect framework in which to reason.
Consider set \begin{align*}P\end{align*} and set \begin{align*}Q\end{align*} that are just collections of things represented by circles. If something is not in the set, then it is not in the circle.
In this case set \begin{align*}P\end{align*} is a subset of set \begin{align*}Q\end{align*} since it is entirely included within set \begin{align*}Q\end{align*}. Mathematically we write the statement “\begin{align*}P\end{align*} is a subset of \begin{align*}Q\end{align*}” as:
\begin{align*}P \subseteq Q\end{align*}
This can be translated to an if-then statement, and simplified using symbols:
If it is an element in \begin{align*}P\end{align*}, then it is an element in \begin{align*}Q\end{align*}.
If \begin{align*}P\end{align*}, then \begin{align*}Q\end{align*}.
\begin{align*}P\rightarrow Q\end{align*}
If-then statements are examples of conditional statements. Sometimes conditional statements are written without an “if” or a “then”, but can be rewritten. The “if” part of the statement (represented by \begin{align*}P\end{align*} above) is called the hypothesis, antecedent or protasis. The “then” part of the statement (represented by \begin{align*}Q\end{align*} above) is called the conclusion, consequent or apodosis.
In order to precisely define the truth value of a conditional statement, we need to consider the four different combinations of the truth value for \begin{align*}P\end{align*} and \begin{align*}Q\end{align*} in relation to the diagram
- If \begin{align*}P\end{align*} is true, then \begin{align*}Q\end{align*} is true. This statement is true because if an object is inside circle \begin{align*}P\end{align*}, then it is definitely inside circle \begin{align*}Q\end{align*}.
- If \begin{align*}P\end{align*} is true, then \begin{align*}Q\end{align*} is false. This statement is false because there is no possible way an object could be inside circle \begin{align*}P\end{align*} and yet outside circle \begin{align*}Q\end{align*}.
- If \begin{align*}P\end{align*} is false, then \begin{align*}Q\end{align*} is true. This statement is considered true because if an object is outside circle \begin{align*}P\end{align*} then it may or may not be in circle \begin{align*}Q\end{align*}. There is no contradiction.
- If \begin{align*}P\end{align*} is false, then \begin{align*}Q\end{align*} is false. This statement is also considered true because if an object is outside circle \begin{align*}P\end{align*}, then it can be outside circle \begin{align*}Q\end{align*}. Like the previous statement, there is no contradiction.
The truth values of the four combinations can be summarized in a truth table. Recall that truth tables are extremely useful for summarizing complicated logical sentences and identifying whether the statements are true or false.
\begin{align*}P\end{align*} | \begin{align*}Q\end{align*} | \begin{align*}P \rightarrow Q\end{align*} |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Note that a conditional statement is only false when the hypothesis is true and the conclusion is false. Also note that any conditional statement with a false hypothesis is trivially true. The following statement is trivially true because the hypothesis is false.
If pigs can fly then butterflies eat elephants.
The truth of this statement confuses many people the first time they look at it. One way to frame it in your mind is to realize that a statement is false only when it results in a logical contradiction. In a world where pigs could fly perhaps butterflies could eat elephants, who knows? It would be ridiculous for a person to argue that in the hypothetical world where pigs could fly that there is no way that butterflies could eat elephants.
Example A
Rewrite the following conditional statements in if-then form.
- If you go to the show, you will be amazed.
- Unless you buy firewood you will be cold.
- Come here and you will get a present.
- Kicking a soccer ball makes it bounce.
- Give me your lunch money or I’ll put you in a locker.
- Anyone who wears orange likes Halloween.
- Without my sunglasses on I can’t drive.
- Buy this product and you’ll be beautiful and popular.
Solution: Even though these statements have words like “and”, “or” and “not” they are still just conditional statements. In each case, consider which action or event leads to another action or event.
- If you go to the show, then you will be amazed.
- If you do not buy firewood, then you will be cold.
- If you come here, then you will get a present.
- If you kick a soccer ball, then it will bounce.
- If you don’t give me your lunch money, then I’ll put you in a locker.
- If a person wears orange, then that person likes Halloween.
- If I do not wear my sunglasses, then I can’t drive.
- If you buy this product, then you will be beautiful and popular.
Example B
Evaluate the truth value of the following conditional statement using a truth table.
If when I go somewhere I always run, then when I run I always go somewhere.
Solution: This statement actually has two layers of conditional statements because both the hypothesis and the conclusion are conditional statements themselves.
- Let \begin{align*}R\end{align*} be the statement “I run”.
- Let \begin{align*}S\end{align*} be the statement “I go somewhere”.
The original sentence can be rewritten in symbols as: \begin{align*}(S\rightarrow R) \rightarrow (R \rightarrow S)\end{align*}. To make the truth table for the sentence, start with all possible truth combinations of \begin{align*}S\end{align*} and \begin{align*}R\end{align*} (both true, \begin{align*}S\end{align*} true/\begin{align*}R\end{align*} false, \begin{align*}S\end{align*} false/\begin{align*}R\end{align*} true, both false). Then, find the truth values for each piece working up to the full sentence.
\begin{align*}S\end{align*} | \begin{align*}R\end{align*} | \begin{align*}S\rightarrow R\end{align*} | \begin{align*}R \rightarrow S\end{align*} | \begin{align*}(S\rightarrow R) \rightarrow (R \rightarrow S)\end{align*} |
T | T | T | T | T |
T | F | F | T | T |
F | T | T | F | F |
F | F | T | T | T |
The statement is always true unless you run but do not go anywhere.
Example C
One day a student went to her logic professor and said, “I don’t believe that when the hypothesis of a conditional statement is true, then the whole statement is trivially true”.
The professor replied, “Alright then. Give me any false hypothesis you can think of and I will mathematically prove that you are a goat.”
The student thought about it and said, “Okay, prove the statement: ‘If \begin{align*}1=2\end{align*} then I am a goat’.”
How would you prove this ridiculous (but true!) statement?
Solution: If \begin{align*}1=2\end{align*}, then any group of 2 things must also be a group of just 1 thing. Therefore in the two element set containing the student and the goat, there must be only one element. Thus, the goat and the student must be the same thing. ∴
Concept Problem Revisited
Use a diagram to represent the conditional statement.
If it rains today then you will stay at home and read a book.
- \begin{align*}P=\ it \ rains \ today\end{align*}
- \begin{align*}Q=\ you \ will \ stay \ at \ home \ and \ read\ a \ book\end{align*}.
You stayed at home and read a book. This implies that \begin{align*}Q\end{align*} is true. According to the diagram, if an object is inside \begin{align*}Q\end{align*} it may or may not be inside \begin{align*}P\end{align*}. Thus, you can conclude nothing about the rain. Many people will want to incorrectly conclude that it must have rained, but conditional statements only flow in one direction.
Vocabulary
A set is a collection of numbers, letters or anything.
A subset is a collection of objects within a larger set.
Set theory studies the relationships of sets and subsets.
A conditional statement is a statement that can be written as an if-then statement.
The “if” part of an if-then statement is called the hypothesis, antecedent or protasis.
The “then” part of an if-then statement is called the conclusion, consequent or apodosis.
Guided Practice
1. Consider the four possible scenarios of the following statement and explain why there is only one way the statement is false.
If Brian is promised cookies then Brian will eat cookies soon.
2. If all of the following statements are true, what can you conclude?
- All babies are cute.
- Laura likes cute people.
- Laura is a baby.
3. There are 53 people in the marching band and 49 people on the swim team. If 84 people belong to either or both teams, how many people are on both teams?
Answers:
1. Scenario \begin{align*}\text A\end{align*} – both hypothesis and conclusion are true. Brian is promised cookies and then later eats cookies. Life is as it should be so the statement is true.
Scenario \begin{align*}B\end{align*} – hypothesis is true but the conclusion is false. Brian is promised cookies, but he does not end up eating cookies. Life is tough for Brian. The statement is false in this case.
Scenario \begin{align*}C\end{align*} – hypothesis is false but the conclusion is true. Brian is never promised cookies, but he’s lucky and gets to eat some anyways. The statement is true.
Scenario \begin{align*}D\end{align*} – hypothesis is false and the conclusion is false. Brian is never promised cookies and never gets any, so the promise isn’t broken. The statement is true.
2. First translate each of the statements into conditional statements (even if they sound awkward!). This is helpful for determining an if-then chain of events.
- \begin{align*}A\end{align*}: If a person is a baby then the person is cute.
- \begin{align*}B\end{align*}: If a person is cute then Laura likes that person.
- \begin{align*}C\end{align*}: If a person is named Laura then that person is a baby.
You should notice the circular structure of these three statements.
\begin{align*}A \rightarrow B, B \rightarrow C, C \rightarrow A\end{align*}
\begin{align*}A \rightarrow B \rightarrow C \rightarrow A\end{align*}
While many conclusions could be made, one conclusion about Laura is that she likes herself.
3. While this problem is not specific to if-then statements, it can be solved using a set theory representation and if-then logic.
Students on both teams would be double counted if you simply added up the number of students in band and the number of students on the swim team.
\begin{align*}53+49=102\end{align*}
Since there are only 84 people total, then \begin{align*}102-84=18\end{align*} students must have been counted twice. Therefore, there are 18 students on both squads.
Practice
1. What are the three names for the “if” part of an if-then statement?
2. What are the three names for the “then” part of an if-then statement?
Rewrite each of the following statements in if-then form.
3. If you like Pepsi, you will like Coke.
4. Do your homework and you will get candy.
5. Anyone who goes to the mall likes to shop.
6. Unless you cook dinner, you will be hungry.
7. Join this program and you will lose weight.
8. Shoveling snow makes your back sore.
9. Without knowing how to drive, you will not get your license.
10. Be nice to your sister or you will be punished.
11. When is the following statement false? If you are a kid, then you like pizza.
12. If all of the following statements are true, what can you conclude?
- All kids like pizza.
- Sam is 8 years old.
- 8 year olds are kids.
13. There are 15 people in the math club and 27 people in the debate club. If 33 people belong to either or both clubs, how many people are just in the debate club?
14. There are 32 people on the bus in the morning and 40 people on the bus in the afternoon. If 20 people only ride the bus in the afternoon, how many people only ride the bus in the morning?
15. There are 24 people in your math class and 27 people in your English class. If 7 people are in your English class but not your math class, how many people are in your math class but not your English class?