# Conditional Sets

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**1. Conditional Sets**

Clues given in such problems are all conditional (“if…then”) in nature. In such type of sets hypothetical symbols can be a big help to comprehend and solve questions.

**Example 1:**

If A is on the team, B must also be on the team.

We can write with the symbol: A → B

Note that this does not mean B → A (i.e. it does not follow that if B is on the team, A has to be in the team)

However, Not B → Not A (i.e. If B is not on the team, A is also not on the team).

This is called a contrapositive deduction.

**Example 2:**

If C is selected, D cannot be selected.

We can use the symbol: C → Not D.

The contrapositive deduction would be: D → Not C (i.e. if D is selected, then C cannot be selected.)

The student has to develop logic for solving these types of questions. Logic is defined as the study of methods and principles used to distinguish good (correct) reasoning from bad (incorrect) reasoning.

# Related Terms

Since most questions in Reasoning are based on whether the student is capable of testing the validity of an Argument, the first thing one has to clearly understand is the concept of the Argument. For the purposes of understanding the concept of the Argument fully, it would help to get acquainted with a few key related terms:

**Propositions**

Propositions are basically the units of an Argument. A typical proposition has a relationship spelled out between a subject and a predicate in the form of a simple (usually) sentence e.g.- All thieves are criminals.

**Conclusions**

The Conclusion of an Argument is the final proposition that is affirmed on the basis of other propositions of the same Arguments, and leading to the same. So basically the conclusion is also a proposition, but one that is derived from another.

**Premise**

The term Premise is applied to that proposition that give rise to the conclusion i.e. that occurs first in sequence and has to be valid for the conclusion to be valid too. For the purposes of explanation, frequently the terms ‘Proposition’ and ‘Premise’ are used interchangeably. So the student should not get confused.

**Inference**

The term Inference is applied to that proposition that is a result of or derivation from the Premiss. So, basically it means the same as Conclusions but is used usually in cases where there is only one premiss from which it is derived.

# Solved Example

A track coach is deciding which and how many of her athletes—L, M, N, 0, P, R, and S—will compete in an upcoming track meet. She will decide according to the following guidelines:

*If L competes, M must compete.**If M and N both compete, O cannot compete.**If N and O both compete, R cannot compete.*-
*If O competes, either P or S must compete.* *Either P or R must compete, but they cannot both compete.**P and S cannot both compete.*

**First step: Symbolize the rules:**

1. L → M, Not M → Not L

2. M & N → Not O, O → Not (M & N)

3. O & N → Not R, R → Not (O & N)

4. O → P / S, Not P / Not S → Not O

5. P → Not R, R → Not P, Either must

6. P → Not S, S → Not P

If only three athletes can compete in the track meet, which of the following could be that group of athletes?

(A) L, M, and N

(B) M, P, and S

(C) M, P, and R

(D) N, O, and P

(E) N, O, and R

Here let us take each option and check for any of the rule violations.

Rule out option A. Rule 5 - Either P/R must. Not present in option A.

Option B violates Rule 6 - P → Not S

Option C violates Rule 5. P → Not R, R → Not P

Option D seems to be correct.

Option E violates rule 3. O & N → Not R

If O and S both compete in the track meet, which of the following must be true?

(A) N competes.

(B) P competes.

(C) R competes.

(D) L does not compete.

(E) M does not compete.

Here try to look immediately at the rules involving people mentioned in the question. What are these rules? Rule 4 – which seems to be satisfied, because either of P or S is included along with O. Next, would be Rule 6, If S is competing, P cannot compete. But rule 5, states that if P is not competing R must compete. So answer is option C.

# Solved Example

If only three athletes can compete in the track meet, which of the following could be that group of athletes?

(A) L, M, and N

(B) M, P, and S

(C) M, P, and R

(D) N, O, and P

(E) N, O, and R

Here let us take each option and check for any of the rule violations.

Rule out option A. Rule 5 - Either P/R must. Not present in option A.

Option B violates Rule 6 - P → Not S

Option C violates Rule 5. P → Not R, R → Not P

Option D seems to be correct.

Option E violates rule 3. O & N → Not R

2. If O and S both compete in the track meet, which of the following must be true?

(A) N competes.

(B) P competes.

(C) R competes.

(D) L does not compete.

(E) M does not compete.

Here try to look immediately at the rules involving people mentioned in the question. What are these rules? Rule 4 – which seems to be satisfied, because either of P or S is included along with O. Next, would be Rule 6, If S is competing, P cannot compete. But rule 5, states that if P is not competing R must compete. So answer is option C.