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:
![Rendered by QuickLaTeX.com \[\text{R}\to\text{P}\]](http://shsatcurriculum.com/wp-content/ql-cache/quicklatex.com-cf81c9e9821f200eab199075a14097ad_l3.svg)
and
![Rendered by QuickLaTeX.com \[\text{P}\to\text{S}\]](http://shsatcurriculum.com/wp-content/ql-cache/quicklatex.com-65a59b191f6645cfbe1b60c893fe42fc_l3.svg)
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:
![Rendered by QuickLaTeX.com \[\text{R}\to\text{S}\]](http://shsatcurriculum.com/wp-content/ql-cache/quicklatex.com-6479346da7ffda91868648a3e0c37274_l3.svg)
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
Let’s put these statements in symbols.
T = Ron goes to the theater
M = Sandra goes to the mall
C = Anna goes to a concert
So the statements are
and
. The sequence of events is
, and we can skip the middle and get
. Putting this back in English, we get “If Ron goes to the theater, the Anna goes to a concert”.
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
Put the statements into symbolic form.
G= You get good grades
P = You can go to the party
S = You study
So the original statements are
and
. Notice both contain
, so that is what we use to connect them. These events are given out of order. The correct sequence of events is
. We skip the middle and get
. In English, this is “If you don’t study, then you can’t go to the party”.
3) 

Solution
We can chain the two if/thens with the
. We get
as our answer.
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
and
to use the chain rule.
Combining the Contrapositive and the Chain Rule
The chain rule says that if we have two if/thens like
and
, we know
must also be true. If this were an SHSAT,
might show up as the correct answer. However, there are other possible correct answers.
Since
is true, its contrapositive must also be true. The contrapositive would be
. 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
Let S = It is snowing
R = Abby goes running
H = Abby plays hockey
(You may choose different letters)
The statements are
and
.
b) Get the contrapositive of both if/thens.
c) If possible, use the chain rule on the statements.
d) Get the contrapositive of the if/then we got from the chain rule.
Solution
The contrapositive of
is
, or “If Abby plays hockey, then it is snowing”. This if/then must be true, so if you see it as an answer choice, it is the correct answer.
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
First we write out our statements symbolically. Let T = It is Tuesday, L = Ann does laundry and D = Ann walks the dog. Then the first if/then is
and the second is
. We can connect these two if/thens with the chain rule and get
, which means “If it is Tuesday, then Ann walks the dog”. This is not an answer choice, so let’s find the contrapositive. The contrapositive of
is
, or “If Ann does not walk the dog, then it is not Tuesday”. This is choice B, the correct answer!
Wrong Answer Rundown
a) If Ann walks the dog, then it is Tuesday.
Wrong- Inverse of correct answer
b) If Ann does not walk the dog, then it is not Tuesday. Correct Answer
c) If Ann does not do laundry, then she does not walk the dog.
Wrong- Inverse of one of the original if/thens
d) If it is not Tuesday, then Ann does not walk the dog.
Wrong- Converse of correct answer
e) Ann likes to get all her chores done at once. Nonsense- statement is too general
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
First we have to turn these statements into if/thens. “When Daniel goes fishing, he has a sandwich for lunch” is equivalent to “If Daniel goes fishing, then he has a sandwich for lunch”. “Daniel does not have a sandwich for lunch if he does not have juice” is equivalent to “If Daniel does not have juice, then he does not have a sandwich for lunch”.
Now let’s put these statements into symbolic form.
Let
F = Daniel goes fishing
S = Daniel has a sandwich for lunch
J = Daniel has juice
Then the two if/thens are
and
. We can’t use the chain rule on these two if/thens. Let’s get the contrapositive of the second one. The contrapositive of
is
. With this new statement, we can use the chain rule. Using the chain rule on
and
, we get
, or “If Daniel goes fishing, then he has juice”. The correct answer is choice E.
Wrong Answer Rundown
a) Daniel only eats fish sandwiches. Nonsense- not enough information
b) If Daniel does not go fishing, then he does not have a sandwich for lunch.
Wrong- Inverse of one of the original if/thens
c) If Daniel does not have a sandwich for lunch, then he does not have juice.
Wrong- Converse of one of the original if/thens
d) If Daniel does not go fishing, then he does not have juice.
Wrong- Inverse of correct answer
e) If Daniel goes fishing, then he has juice. Correct Answer
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:
and
.
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
is
, or “If Kayla does not join the art club, then she does not join the track team”. The contrapositive of
is
, 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:


Contrapositives:


We can’t use the chain rule with the two original statements. However, we can use it on
and
to get
, 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
and
to get
, 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
is
. The contrapositive of
is
. 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:




Wrong Answer Rundown
A) Kayla does not join the track team if she joins the debate team
Correct Answer
B) Kayla joins the track team if she joins the art club
Wrong- Converse of one of the original if/thens
C) If Kayla does not join the debate team, then she joins the track team
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
Wrong- We don’t know what happens if Kayla doesn’t join the debate team
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