A conditional statement where the truth of one event guarantees the truth of another.
Conditional statements are sometimes called “if/then” statements. They are essential in many branches of mathematics, especially logic.
This section covers:
- What is a Conditional Statement?
- Conditional Statement Definition
- Conditional Statement Examples
What Is a Conditional Statement?
A conditional statement describes a relationship between two events where the truth of one event implies the truth of the other.
If an event, $P$, implies the truth of an event $Q$, then the conditional statement describing the relationship is “If $P$, then $Q$. Alternatively, this is “$P$ implies $Q$.” In logic, this is $P \rightarrow Q$.
A conditional statement can be true or false. A true statement is true no matter what, while a false statement is false in one or more instances. There are examples of this below.
The event on the left side of the arrow is called the antecedent of the statement, while the event on the right side of the arrow is called the consequence of the statement. An antecedent is called a “sufficient condition” because knowing it is true is enough to know the consequence is true (when the statement itself is true). The consequence is called a necessary condition because it is a prerequisite for the antecedent to be true in a true statement.
Conditional statements have converses, inverse, and contrapositives. Specifically, for a statement $P \rightarrow Q$:
Converse: $Q \rightarrow P$
Inverse: $\neg P \rightarrow \neg Q$
Contrapositive: $\neg Q \rightarrow \neg P$.
The truth value of the conditional statement will always be the same as the truth value of the contrapositive statement. Likewise, the truth value of the converse will always be the same as the truth value of the inverse.
Note that just because an event $P$ implies an event $Q$, the event $Q$ does not necessarily mean the event $P$.
A biconditional statement is one describing a relationship between two events where each implies the other. They are written $P \leftrightarrow Q$ and read “$P$ if and only if $Q$.” Sometimes the relationship is also written as $P$ iff. $Q$.
A biconditional statement is true if and only if the statement and its converse are both true.
Conditional Statement Definition
A conditional statement relates two events where the second event depends on the first. This statement can be true or false.
Conditional statements are also called “if/then” statements because if an event $Q$ follows from an event $P$, the conditional statement is “if $P$, then $Q$.”
In formal logic, this is:
“$P \rightarrow Q$.”
This is read:
“$P$ implies $Q$.”
In this case, $P$ is called the antecedent, and $Q$ is called the consequence.
Conditional Statement Examples
Consider the following events:
$P$: An animal is a cat.
$Q$: An animal is a mammal.
$R$: An animal is a fish.
Then, consider the following conditional statements:
- $P \rightarrow Q$
- $Q \rightarrow P$
- $R \rightarrow Q$
All of these are examples of conditional statements. Only one, however, represents a true statement.
$P \rightarrow Q$ is true because if an animal is a cat, it must be a mammal.
But $Q \rightarrow P$ is false. Even though if an animal is a mammal, it may be a cat, or it may not be a cat. If a statement is false in even one circumstance, it is labeled false.
Finally, $R \rightarrow Q$ is always false. No fish are mammals.
Note that the consequences and antecedents can be more complex than this. They can include “and” and “or” operators, for example.
This section covers common examples of problems involving conditional statements and their step-by-step solutions.
Decide whether each mathematical-conditional statement is true or false. If it is false, give at least one specific counterexample.
- If it is a prime number, then it is odd.
- If it is the sum of two even numbers, then it is odd.
- If it is a square, then it is a quadrilateral.
The first statement is false. Every prime number except $2$ is odd. Even though there is only one exception, however, the whole statement is false.
The statement could be amended to “if it is a prime number other than two, then it is odd.” It would be true.
The second statement is always false, but it is only necessary to find one counterexample. $2$ and $4$ are even numbers, but their sum is $6$, which is not odd.
This statement is true by appealing to definitions. Since figures with four sides are quadrilaterals, and a square is a figure with four sides, a square is a quadrilateral.
Give a real-life example to show why the contrapositive of a true statement must be true, but the converse doesn’t have to be.
Consider the third statement from example 1. “If it is a square, then it is a quadrilateral.”
The contrapositive of this statement is “if it is not a quadrilateral, then it is not a square.”
Things that are not quadrilaterals have a number of sides greater than or less than four. Since a square has four sides, the figure that is not a quadrilateral cannot be a square.
But, consider the converse “if it is not a square, then it is not a quadrilateral.” This statement is false because, for example, a rectangle is a quadrilateral that is not a square.
Give a real-life example of a biconditional statement, spelling out the conditional statement, the converse, the inverse, and the contrapositive.
Biconditional statements are usually definitional. A famous example is “it is a right angle if and only if it measures 90 degrees.” This is the conditional statement, and it is true by definition.
Converse: “If it is not a right angle, it is not 90 degrees.” Again, this must be true because 90-degree angles are right angles.
Inverse: “If it is 90 degrees, then it is a right angle.” Once more, the definition holds this to be true.
Contrapositive: “If it is not 90 degrees, then it is not a right angle.” Since the original conditional statement is true, this statement is true too.
Is the following conditional statement true or false?
If $x$ is a real number and $x^2$ is positive, then $x$ is positive.
This statement is false. Again, it is only necessary to find one counterexample, though there are infinitely many.
Let $x=-2$. Then $x$ is a real number, and $x^2=4$ is positive. Yet, $x$ itself is negative. Therefore, the statement is false.
$P \rightarrow Q$ is true.
$Q \rightarrow R$ is true.
Does $P \rightarrow R$? Can you think of events that fit this relationship?
In this case, yes.
$P \rightarrow R$ means that if $P$ is true, then $R$ must be true.
If $P$ is true, $Q$ must be true by the first conditional statement. Similarly, if $Q$ is true, then $R$ must be true by the second conditional statement.
Thus, if $P$ is true, then $Q$ is true, and then $R$ is true. Thus, $P \rightarrow R$ is true.
Consider this example. Let $P$ be the event “it is a cat,” $Q$ be the event “it is a feline,” and $R$ be the event “it is a mammal.”
If an animal is a cat, it is a feline. If an animal is a feline, it is a mammal. Similarly, all cats are mammals.
Note, however, that not all felines are cats (at least, not house cats). Some are tigers or lions. Likewise, not all mammals are felines. Some are primates, and others are canines, etc.
- Given that the contrapositive of a true statement is always true, prove that if the converse of a statement is true, then so is the inverse.
- Does $\neg (P \cap Q) \rightarrow (\neg P \cap \neg Q)$?
- Does $\neg (P \cap Q) \rightarrow (\neg P \cup \neg Q)$?
- Is the statement “If it has three 60-degree angles, then it is an equilateral triangle” true? How could it be amended?
- Prove that the statement “If it is divisible by 3, then it is odd” is false.
- Consider a conditional statement $P \rightarrow Q$. If this statement is true, then the contrapositive $\neg Q \rightarrow \neg P$ is also true.
Now, assume the converse, $Q \rightarrow P$ is true. The contrapositive of this statement is $\neg P \rightarrow \neg Q$. This, however, is the inverse of the original statement. Thus, a true converse implies a true inverse because the inverse is the contrapositive of the converse.
- No. $\neg (P \cap Q)$ is “not $P$ and $Q$. That is, both are not true. Therefore, at least one is false. However, $\neg P \cap \neg Q$ means both are false. Therefore, $\neg P \cap Q$ is a counterexample.
- This is true. As before, $\neg (P \cap Q)$ means at least one of $P$ or $Q$ is false. That is, $\neg P \cup \neg Q$. Note that this includes the case where both are false, the case where $P$ is false, and $Q$ is true, and the case where $Q$ is true and $P$ is false.
- At face value, this statement looks true. $60+60+60=180$. However, there is no stipulation that the figure with three 60-degree angles only has three sides. For example, it could be a pentagon with three 60-degree sides and two other sides that add up to 360 (this would have to be a concave pentagon). Amending this statement to “If it has only three sides and each side measures 60-degrees, then it is an equilateral triangle” makes it unquestionably true.
- It is only necessary to find one counterexample. Consider the number $6$, which is both even and divisible by $3$.