# 2.2: Conditional Statements

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Identify the hypothesis and conclusion of an if-then statement.
• Write the converse, inverse, and contrapositive of an if-then statement.

## Review Queue

Find the next figure or term in the pattern.

1. 5, 8, 12, 17, 23, \begin{align*}\ldots\end{align*}
2. \begin{align*}\frac{2}{5},\frac{3}{6},\frac{4}{7},\frac{5}{9},\frac{6}{10}, \ldots\end{align*}
3. Find a counterexample for the following conjectures.
1. If it is April, then it is Spring Break.
2. If it is June, then I am graduating.

Know What? Rube Goldman was a cartoonist in the 1940s who drew crazy inventions to do very simple things. The invention to the right has a series of smaller tasks that leads to the machine wiping the man’s face with a napkin.

Describe each step, from \begin{align*}A\end{align*} to \begin{align*}M\end{align*}.

## If-Then Statements

Conditional Statement (also called an If-Then Statement): A statement with a hypothesis followed by a conclusion.

Hypothesis: The first, or “if,” part of a conditional statement.

Conclusion: The second, or “then,” part of a conditional statement. The conclusion is the result of a hypothesis.

If-then statements might not always be written in the “if-then” form.

Statement 1: If you work overtime, then you’ll be paid time-and-a-half.

Statement 2: I’ll wash the car if the weather is nice.

Statement 3: If 2 divides evenly into \begin{align*}x\end{align*}, then \begin{align*}x\end{align*} is an even number.

Statement 4: I’ll be a millionaire when I win monopoly.

Statement 5: All equiangular triangles are equilateral.

Statements 1 and 3 are written in the “if-then” form. The hypothesis of Statement 1 is “you work overtime.” The conclusion is “you’ll be paid time-and-a-half.”

So, if Sarah works overtime, then what will happen? From Statement 1, we can conclude that she will be paid time-and-a-half.

If 2 goes evenly into 16, what can you conclude? From Statement 3, we know that 16 must be an even number.

Statement 2 has the hypothesis after the conclusion. If the word “if” is in the middle of the statement, then the hypothesis is after it. The statement can be rewritten:

If the weather is nice, then I will wash the car.

Statement 4 uses the word “when” instead of “if” and is like Statement 2. It can be written:

If I win monopoly, then I will be a millionaire.

Statement 5 “if” and “then” are not there. It can be rewritten:

If a triangle is equiangular, then it is equilateral.

Example 1: Use the statement: I will graduate when I pass Calculus.

a) Rewrite in if-then form.

b) Determine the hypothesis and conclusion.

Solution: This statement is like Statement 4 above. It should be:

If I pass Calculus, then I will graduate.

The hypothesis is “I pass Calculus,” and the conclusion is “I will graduate.”

## Converse, Inverse, and Contrapositive

Look at Statement 2 again: If the weather is nice, then I’ll wash the car.

This can be rewritten using letters to represent the hypothesis and conclusion.

\begin{align*}p = \text{the weather is nice} && q = \text{I’ll wash the car}\end{align*}

Now the statement is: If \begin{align*}p\end{align*}, then \begin{align*}q\end{align*}.

An arrow can also be used in place of the “if-then”: \begin{align*}p \rightarrow q\end{align*}

We can also make the negations, or “nots” of \begin{align*}p\end{align*} and \begin{align*}q\end{align*}. The symbolic version of not \begin{align*}p\end{align*}, is \begin{align*}\sim p\end{align*}.

\begin{align*}\sim p = \text{the weather is not nice} && \sim q = \text{I won’t wash the car}\end{align*}

Using these “nots” and switching the order of \begin{align*}p\end{align*} and \begin{align*}q\end{align*}, we can create three new statements.

\begin{align*}& \text{Converse} && q \rightarrow p && \underbrace{\text{If I wash the car}}_{q}, \ \underbrace{\text{then the weather is nice}}_{p}.\\ & \text{Inverse} && \sim p \rightarrow \sim q && \underbrace{\text{If the weather is not nice}}_{\sim p}, \ \underbrace{\text{then I won't wash the car}}_{\sim q}.\\ & \text{Contrapositive} && \sim q \rightarrow \sim p && \underbrace{\text{If I don't wash the car}}_{\sim q}, \ \underbrace{\text{then the weather is not nice}}_{\sim p}.\end{align*}

If the “if-then” statement is true, then the contrapositive is also true. The contrapositive is logically equivalent to the original statement. The converse and inverse may or may not be true.

Example 2: If \begin{align*}n > 2\end{align*}, then \begin{align*}n^2 > 4\end{align*}.

a) Find the converse, inverse, and contrapositive.

b) Determine if the statements from part a are true or false. If they are false, find a counterexample.

Solution: The original statement is true.

\begin{align*}& \underline{\text{Converse}}: && \text{If} \ n^2 > 4, \ \text{then} \ n > 2. && False. \ \text{If} \ n^2 = 9, n = -3 \ \text{or} \ 3. \ (-3)^2=9\\ & \underline{\text{Inverse}}: && \text{If} \ n < 2, \ \text{then} \ n^2 < 4. && False. \ \text{If} \ n=-3 , \ \text{then} \ n^2=9.\\ & \underline{\text{Contrapositive}}: && \text{If} \ n^2 < 4, \ \text{then} \ n < 2. && True. \ \text{the only} \ n^2 < 4 \ \text{is} \ 1. \ \sqrt{1}=\pm 1\\ & &&&& \quad \qquad \text{which are both less then 2.} \end{align*}

Example 3: If I am at Disneyland, then I am in California.

a) Find the converse, inverse, and contrapositive.

b) Determine if the statements from part a are true or false. If they are false, find a counterexample.

Solution: The original statement is true.

\begin{align*}& \underline{\text{Converse}}: && \text{If I am in California, then I am at Disneyland.}\\ &&& False. \ \text{I could be in San Francisco.}\\ & \underline{\text{Inverse}}: && \text{If I am not at Disneyland, then I am not in California.}\\ &&& False. \ \text{Again, I could be in San Francisco.}\\ & \underline{\text{Contrapositive}}: && \text{If I am not in California, then I am not at Disneyland.}\\ &&& True. \ \text{If I am not in the state, I couldn’t be at Disneyland.}\end{align*}

Notice for the converse and inverse we can use the same counterexample.

Example 4: Any two points are collinear.

a) Find the converse, inverse, and contrapositive.

b) Determine if the statements from part a are true or false. If they are false, find a counterexample.

Solution: First, change the statement into an “if-then” statement:

If two points are on the same line, then they are collinear.

\begin{align*}& \underline{\text{Converse}}: && \text{If two points are collinear, then they are on the same line.} \ True.\\ & \underline{\text{Inverse}}: && \text{If two points are not on the same line, then they are not collinear.} \ True.\\ & \underline{\text{Contrapositive}}: && \text{If two points are not collinear, then they do not lie on the same line}. \ True.\end{align*}

## Biconditional Statements

Example 4 is an example of a biconditional statement.

Biconditional Statement: When the original statement and converse are both true.

\begin{align*}p \rightarrow q\end{align*} is true

\begin{align*}q \rightarrow p\end{align*} is true

then, \begin{align*}p \leftrightarrow q\end{align*}, said “\begin{align*}p\end{align*} if and only if \begin{align*}q\end{align*}

Example 5: Rewrite Example 4 as a biconditional statement.

Solution: If two points are on the same line, then they are collinear can be rewritten as:

Two points are on the same line if and only if they are collinear.

Replace the “if-then” with “if and only if” in the middle of the statement.

Example 6: The following is a true statement:

\begin{align*}m\angle ABC > 90^\circ\end{align*} if and only if \begin{align*}\angle ABC\end{align*} is an obtuse angle.

Determine the two true statements within this biconditional.

Solution: Statement 1: If \begin{align*}m \angle ABC > 90^\circ\end{align*}, then \begin{align*}\angle ABC\end{align*} is an obtuse angle.

Statement 2: If \begin{align*}\angle ABC\end{align*} is an obtuse angle, then \begin{align*}m\angle ABC > 90^\circ\end{align*}.

This is the definition of an obtuse angle. All geometric definitions are biconditional statements.

Example 7: \begin{align*}p: x < 10 \quad q: 2x < 50\end{align*}

a) Is \begin{align*}p \rightarrow q\end{align*} true? If not, find a counterexample.

b) Is \begin{align*}q \rightarrow p\end{align*} true? If not, find a counterexample.

c) Is \begin{align*}\sim p \rightarrow \sim q\end{align*} true? If not, find a counterexample.

d) Is \begin{align*}\sim q \rightarrow \sim p\end{align*} true? If not, find a counterexample.

Solution:

a) If \begin{align*}x < 10\end{align*}, then \begin{align*}2x < 50\end{align*}. True.

b) If \begin{align*}2x < 50\end{align*}, then \begin{align*}x < 10\end{align*}. False, \begin{align*}x = 15\end{align*}

c) If \begin{align*}x > 10\end{align*}, then \begin{align*}2x > 50\end{align*}. False, \begin{align*}x = 15\end{align*}

d) If \begin{align*}2x > 50\end{align*}, then \begin{align*}x > 10\end{align*}. True, \begin{align*} x \ge 25\end{align*}

Know What? Revisited The series of events is as follows:

If the man raises his spoon, then it pulls a string, which tugs the spoon back, then it throws a cracker into the air, the bird will eat it and turns the pedestal. Then the water tips over, which goes into the bucket, pulls down the string, the string opens the box, where a fire lights the rocket and goes off. This allows the hook to pull the string and then the man’s face is wiped with the napkin.

## Review Questions

• Questions 1-6 are similar to Statements 1-5 and Example 1.
• Questions 7-16 are similar to Examples 2, 3, and 4.
• Questions 17-22 are similar to Examples 5 and 6.
• Questions 23-25 are similar to Example 7.

For questions 1-6, determine the hypothesis and the conclusion.

1. If 5 divides evenly into \begin{align*}x\end{align*}, then \begin{align*}x\end{align*} ends in 0 or 5.
2. If a triangle has three congruent sides, it is an equilateral triangle.
3. Three points are coplanar if they all lie in the same plane.
4. If \begin{align*}x = 3\end{align*}, then \begin{align*}x^2 = 9\end{align*}.
5. If you take yoga, then you are relaxed.
6. All baseball players wear hats.
7. Write the converse, inverse, and contrapositive of #1. Determine if they are true or false. If they are false, find a counterexample.
8. Write the converse, inverse, and contrapositive of #5. Determine if they are true or false. If they are false, find a counterexample.
9. Write the converse, inverse, and contrapositive of #6. Determine if they are true or false. If they are false, find a counterexample.
10. Find the converse of #2. If it is true, write the biconditional of the statement.
11. Find the converse of #3. If it is true, write the biconditional of the statement.
12. Find the converse of #4. If it is true, write the biconditional of the statement.

For questions 13-16, use the statement:

If \begin{align*}AB = 5\end{align*} and \begin{align*}BC = 5\end{align*}, then \begin{align*}B\end{align*} is the midpoint of \begin{align*}\overline{AC}\end{align*}.

1. Is this a true statement? If not, what is a counterexample?
2. Find the converse of this statement. Is it true?
3. Find the inverse of this statement. Is it true?
4. Find the contrapositive of #14. Which statement is it the same as?

Find the converse of each true if-then statement. If the converse is true, write the biconditional statement.

1. An acute angle is less than \begin{align*}90^\circ\end{align*}.
2. If you are at the beach, then you are sun burnt.
3. If \begin{align*}x > 4\end{align*}, then \begin{align*}x+3>7\end{align*}.

For questions 20-22, determine the two true conditional statements from the given biconditional statements.

1. A U.S. citizen can vote if and only if he or she is 18 or more years old.
2. A whole number is prime if and only if its factors are 1 and itself.
3. \begin{align*}2x = 18\end{align*} if and only if \begin{align*}x = 9\end{align*}.

For questions 23-25, determine if:

(a) \begin{align*}p \rightarrow q\end{align*} is true.

(b) \begin{align*}q \rightarrow p\end{align*} is true.

(c) \begin{align*}\sim p \rightarrow \sim q\end{align*} is true.

(d) \begin{align*}\sim q \rightarrow \sim p\end{align*} is true.

If any are false, find a counterexample.

1. \begin{align*}p:\end{align*} Joe is 16. - \begin{align*}q:\end{align*} He has a driver’s license.
2. \begin{align*}p:\end{align*} A number ends in 5. - \begin{align*}q:\end{align*} It is divisible by 5.
3. \begin{align*}p:\end{align*} \begin{align*}x = 4\end{align*} - \begin{align*}q:\end{align*} \begin{align*}x^2 = 16\end{align*}

1. 30
2. \begin{align*}\frac{7}{11}\end{align*}
1. It could be another day that isn’t during Spring Break. Spring Break doesn’t last the entire month.
2. You could be a freshman, sophomore or junior. There are several counterexamples.

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