If / then questions part 2

In part 1 we learned how to solve basic if/then questions. A few key points: Put all statements into if/then form. The contrapositive of a true if/then must also be true. The converse and inverse do not have to be true and are often wrong answer choices. Now we will learn some techniques for dealing with trickier questions where you are given more than one if/then.

The Chain Rule

When we have two if/then statements, sometimes we can connect them. For example, suppose these two statements are true:

If I go to Roma’s restaurant, then I get pizza. If I get pizza, then I get a soda.

This is a chain of events: when I go to Roma’s, I get pizza, and when I get pizza I also get a soda. So whenever I go to Roma’s I end up getting a soda! We can get a new if/then by skipping over the middle step: If I go to Roma’s restaurant, then I get a soda. This must be true!

Lets look at this problem in symbols. Let R = “I go to Roma’s restaurant”, P = “I get pizza”, and S = “I get a soda”. Then our statements are:

$\text{R}\to\text{P}$

and

$\text{P}\to\text{S}$

R leads to P and P leads to S, so R actually leads to S. The Chain Rule lets us get the new if/then:

$\text{R}\to\text{S}$

Let’s try using the chain rule on a few examples:

1) If Ron goes to the theater, then Sandra goes to the mall. If Sandra goes to the mall, then Anna goes to a concert.

Solution

2) If you don’t get good grades, then you can’t go to the party. If you don’t study then you don’t get good grades.

Solution

3) $\text{P}\to\sim\text{Q}$

$\sim\text{Q}\to\text{R}$

Solution

An important note: you can only use the chain rule on two statements if the “then” part of one of the statements is the “if” part of the other. For example, we can use the chain rule on “If Jane goes to the cafeteria, then Bob goes to the cafeteria” and “If it is Tuesday, then Jane goes to the cafeteria” because “Jane goes to the cafeteria” if the “then” part of the second if/then and the “if” part of the first. We can’t use the chain rule with “If an alien is from planet Neon, then they eat florks” and “If an alien has a flying saucer, then they eat florks” because the part they share, “they eat florks”, is the “then” part of both if/thens.

In symbols, if our 3 statements are A, B and C, our statements need to look like $\text{A}\to\text{B}$ and $\text{B}\to\text{C}$ to use the chain rule.

Combining the Contrapositive and the Chain Rule

The chain rule says that if we have two if/thens like $\text{P}\to\text{Q}$ and $\text{Q}\to\text{R}$ , we know $\text{P}\to\text{R}$ must also be true. If this were an SHSAT, $\text{P}\to\text{R}$ might show up as the correct answer. However, there are other possible correct answers.

Since $\text{P}\to\text{R}$ is true, its contrapositive must also be true. The contrapositive would be $\sim\text{R}\to\sim\text{P}$ . So using the chain rule and then taking the contrapositive would be another possible correct answer.

The other possible correct answers you might see are just the contrapositives of the original statements.

Let’s take two if/thens and write out all the possible answers:

If it is not snowing, then Abby goes running. If Abby goes running, then she does not play hockey.

a) Write both statements symbolically.

Solution

b) Get the contrapositive of both if/thens.

Solution

c) If possible, use the chain rule on the statements.

Solution

d) Get the contrapositive of the if/then we got from the chain rule.

Solution

Now for a few SHSAT-style questions:

4) Suppose the following statements are true: If it is Tuesday, then Ann does laundry. If Ann does laundry, then she walks the dog. Which of the following must also be true?

a) If Ann walks the dog, then it is Tuesday.

b) If Ann does not walk the dog, then it is not Tuesday.

c) If Ann does not do laundry, then she does not walk the dog.

d) If it is not Tuesday, then Ann does not walk the dog.

e) Ann likes to get all her chores done at once.

Solution

5) Suppose the following statements are true: When Daniel goes fishing, he has a sandwich for lunch. Daniel does not have a sandwich for lunch if he does not have juice. Which of the following must also be true?

a) Daniel only eats fish sandwiches.

b) If Daniel does not go fishing, then he does not have a sandwich for lunch.

c) If Daniel does not have a sandwich for lunch, then he does not have juice.

d) If Daniel does not go fishing, then he does not have juice.

e) If Daniel goes fishing, then he has juice.

Solution

6) Suppose the following statements are true: If Kayla joins the track team, then she joins the art club. If Kayla joins the debate team, then she does not join the art club. Which of the following must also be true?

A) Kayla does not join the track team if she joins the debate team

B) Kayla joins the track team if she joins the art club

C) If Kayla does not join the debate team, then she joins the track team

D) Kayla is the busiest kid in school

E) If Kayla does not join the debate team, then she does not join the art club

Solution

First let’s write these statements symbolically.

T = Kayla joins the track team

A = Kayla joins the art club

D = Kayla joins the debate team

So the if/thens are: $\text{T}\to\text{A}$ and $\text{D}\to\sim\text{A}$ .

To solve this one, let’s go through all the possible answers we can get from these two if/thens.

i) Contrapositives

The contrapositive of $\text{T}\to\text{A}$ is $\sim\text{A}\to\sim\text{T}$ , or “If Kayla does not join the art club, then she does not join the track team”. The contrapositive of $\text{D}\to\sim\text{A}$ is $\text{A}\to\sim\text{D}$ , or “If Kayla joins the art club, then she does not join the debate team”. Neither of these contrapositives are answer choices in this particular problem.

ii) Chain rule

Here are all the if/thens we know so far:

Original if/thens:

$\text{T}\to\text{A}$

$\text{D}\to\sim\text{A}$

Contrapositives:

$\sim\text{A}\to\sim\text{T}$

$\text{A}\to\sim\text{D}$

We can’t use the chain rule with the two original statements. However, we can use it on $\text{T}\to\text{A}$ and $\text{A}\to\sim\text{D}$ to get $\text{T}\to\sim\text{D}$ , or “If Kayla joins the track team, then she does not join the debate team”. This is not an answer choice. We can also use the chain rule on $\text{D}\to\sim\text{A}$ and $\sim\text{A}\to\sim\text{T}$ to get $\text{D}\to\sim\text{T}$ , or “If Kayla joins the debate team, then she does not join the track team”. This is equivalent to answer A, “Kayla does not join the track team if she joins the debate team”! Remember to put everything in if/then form!

iii) Contrapositive of chain rule

Let’s get the contrapositives of the two if/thens we got from the chain rule. The contrapositive of $\text{T}\to\sim\text{D}$ is $\text{D}\to\sim\text{T}$ . The contrapositive of $\text{D}\to\sim\text{T}$ is $\text{T}\to\sim\text{D}$ . So they are contrapositives of each other! In this case, this step does not give us any new if/thens.

So, here is a list of all the conclusions we can make that would be correct answer choices:

$\sim\text{A}\to\sim\text{T}$

$\text{A}\to\sim\text{D}$

$\text{T}\to\sim\text{D}$

$\text{D}\to\sim\text{T}$

Wrong Answer Rundown

A) Kayla does not join the track team if she joins the debate team $\text{D}\to\sim\text{T}$ Correct Answer

B) Kayla joins the track team if she joins the art club $\text{A}\to\text{T}$ Wrong- Converse of one of the original if/thens

C) If Kayla does not join the debate team, then she joins the track team $\sim\text{D}\to\text{T}$ Wrong- Inverse of correct answer

D) Kayla is the busiest kid in school Nonsense- not enough information

E) If Kayla does not join the debate team, then she does not join the art club $\sim\text{D}\to\sim\text{A}$ Wrong- We don’t know what happens if Kayla doesn’t join the debate team

SHSAT Curriculum

Tips from an SHSAT tutor

The Chain Rule

Combining the Contrapositive and the Chain Rule

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The Chain Rule

Combining the Contrapositive and the Chain Rule

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