Logical Deduction
Logical Reasoning



1. Introduction

In its simplest form, the logical condition is a statement which states that an event depends on another event. The name ‘logical’ is derived from the fact that the occurrence of the second event depends only on the occurrence or non-occurrence of the first.

As we shall see later, such statements can be of only four different types. We then make certain inferences from the given statement and based on those inferences we answer the questions given.

More complicated problems involve more than one statement of the type given above. They can also involve some element of the linear or complex arrangement.

For example, we may be given a scenario in which we have to select a team of three people out of six candidates. We will be given conditions which state that some of the candidates will not be in the team if certain other candidates are selected. Using the basic skills which we have developed in the first part, we treat each statement separately and extract all the information from it.

We are then able to combine the statements (which have taken the form of a series of conditions) to answer any questions that may be asked.

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2. Conditional Notations

Conditional notations are the symbols used to represent logical statements in a concise manner. Let us consider two statements $A$ and $B$:

$A:$ It is raining

$B:$ Mukesh carries an umbrella

We will now see how these two statements can be connected by symbols and in what way should they be interpreted. The following are the four basic logical connectors


1- OR:

$A \vee B:$ This is read as ‘$A\text{ or }B$’. This expression is evaluated if $A$ or $B$ (either of the events) or $A$ and $B$ (both the events) happen.

To put it simply, this expression is said to be true if either one of the events $A$ or $B$ or both of them are true.

In our example, $A \vee B$ will be true if:

a. It is raining, but Mukesh does not carry an umbrella

b. Mukesh is carrying an umbrella, but it is not raining

c. It is raining and Mukesh is carrying an umbrella

This symbol is also called the ‘OR’ connector


2- AND:

$A \wedge B:$ This is read as ‘$A\text{ and }B$’. This expression is evaluated if and only if both events ($A$ and $B$ in this example) happen.

This means that the expression is said to be true whenever both statements $A$ and $B$ are true.

In our example, $A \wedge B$ is true if:

It is raining and Mukesh is carrying an umbrella.

This symbol is also called the ‘AND’ connector.


3- NOT:

$~A$: This is read as ‘not A’. The meaning is ‘if the event $A$ does not happen’.

This symbol negates whatever statement $A$ stands for. In our example, $~A$ simply means that it is not raining.

Similarly, $~B$ means that Mukesh is not carrying an umbrella.


4- EXOR:

$A \oplus B:$ This is read as ‘$A\text{ exor }B$’. This expression is evaluated if and only if one of the events and not both of them happen.

In simple terms, this expression will be true if any one of the statements $A$ or $B$ is true. It will not be true if neither or both statements are true.

In our example, $A \oplus B$ is true if:

a. It is raining and Mukesh is not carrying an umbrella

b. It is not raining and Mukesh is carrying an umbrella

This symbol is also called the ‘EXOR’ connector.


5- IF, THEN:

$A \rightarrow B:$ This is read as ‘if $A$ happens, then $B$ will happen’.

This simply means that if $A$ is true, then $B$ will have to be true.

In the example, $A \rightarrow B$ will read something like this:

If it is raining, then Mukesh will carry an umbrella.

Logical statements are generally represented by two expressions (using and, or and not operators) separated by an ‘if – then’ sign.

An example of such an expression is ($A \vee B) \wedge ~C \rightarrow D$. This expression means ‘if $A$ or $B$ happens, and $C$ does not happen, then $D$ will happen’.

Now that we have developed the basic symbolism to represent logical statements, let us take a closer look at the statements themselves.

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2.1. IF A, THEN B

This class of statements tells us that an event $B$ will happen if an event $A$ happens before it. Note that it does not tell us anything about the other way round, i.e. if the event $B$ happens, event $A$ may or may not have happened. The only thing the statement does tell us is that, if event $A$ happens, event $B$ is bound to happen.

To explain this concept clearly, let us use the same example that we used in the previous section.

$A \rightarrow B$ means that:

‘If it is raining, then Mukesh will carry an umbrella’


2.1.1. Representation

The representation of this statement is the simplest of all. It is just written as $A \rightarrow B$.

Let us look at the different ways in which this statement can be interpreted:

  • Problems based on this statement are usually easy to solve and mostly revolve around the fact that the relationship is not two-way.
    • This implies that if $A \rightarrow B$ is true, it does not mean that $B \rightarrow A$ is also true.
    • In our example, $A \rightarrow B$ does not imply that:
    • ‘If Mukesh is carrying an umbrella, then it is raining’
  • One way to approach such a problem is to look at all possible cases from the point of view of the first event. For example, imagine that we are told that if $A$ happens, then $B$ will happen. The possible cases are:
    • $A$ happens. In this case, $B$ will happen.
    • $A$ does not happen. In this case, $B$ may or may not happen (since we have been given no statement involving $~A$).
    • In our example, this means that:
    • If it is raining, then Mukesh will carry an umbrella.
    • However, if it is not raining, Mukesh may or may not carry an umbrella.
  • Another way to approach such a problem is to look at all possible cases from the point of view of the second event. If we are told that if $A$ happens, $B$ will happen, then there are two possible cases:
    • $B$ happens. In this case, $A$ may or may not have happened.
    • $B$ does not happen. In this case, $A$ could not have happened since, if $A$ happens, $B$ has to happen.
  • In the example, this means that:
  • If Mukesh is carrying an umbrella, it may or may not be raining.
  • However, if Mukesh is not carrying an umbrella, then it is definitely not raining.
  • Thus, we can see that a reversal is possible, but that it is accompanied by negation. In terms of our notation: if $A \rightarrow B$, then $~B \rightarrow ~A$.

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Example:

If Ram studies, he will pass his exam. Ram fails the exam. Which of the following statements is true?

(1) Ram studied for the exam

(2) Ram did not study for the exam

(3) Ram may or may not have studied for the exam

(4) None of these

Solution:

We look at the problem from the point of view of the second event. Call $A$ the event that Ram studies for the exam. Call $B$ the event that Ram passes the exam. We have been given the information that $A \rightarrow B$.

Thus, $~B \rightarrow ~A$. Putting this in words, we learn that, if Ram failed the exam, he did not study for it.

Hence, option 2 is the correct choice.

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2.2. IF A, THEN NOT B

This class of statements tells us that an event $B$ cannot happen if an event $A$ happens before it. Again, note that it does not tell us anything about the other way round. This means that, if we know that event $B$ has not happened, event $A$ may or may not have happened.

Using our example, $A \rightarrow ~B$ means that:

‘If it is raining, then Mukesh is not carrying an umbrella.’


2.2.1. Representation

This statement is written as $A \rightarrow ~B$.

The reversal of this statement (we saw something similar for the first case) is $B \rightarrow ~A$. Thus, if the occurrence of an event $A$ implies non-occurrence of an event B, then the occurrence of event B implies that event $A$ never happened.

Thus, if $A \rightarrow ~B$ then $B \rightarrow ~A$

Let us look at the different ways in which this statement can be interpreted:


First, method of approach:

looking at all possible cases from the point of view of the first event. We are given that $A \rightarrow ~B$. The possible cases are:

i. $A$ happens: In this case, $B$ will not happen

In our example, this simply means that:

If it is raining, Mukesh is not carrying an umbrella.

ii. $A$ does not happen: In this case, $B$ may or may not happen (since we have been given no statement involving $~A$ on the LHS)

In our example, this means that:

If it is not raining, Mukesh may or may not be carrying an umbrella.


Second method of approach:

Looking at all possible cases from the point of view of the second event.

We are given that $A \rightarrow ~B$, hence we are also given its logical negation, that $B \rightarrow ~A$.

i. $B$ happens: In this case, $A$ could not have happened.

In our example, this simply means that:

‘If Mukesh is carrying an umbrella, then it is definitely not raining’

ii. $B$ does not happen: In this case, $A$ may or may not have happened.

In our example, this means that:

If Mukesh is not carrying an umbrella, then it may or may not be raining.

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Example:

If a thief sees a policeman on a given night, he will not commit a robbery that night. One night, a thief does not see a policeman. Which of the following statements is true?

(1) The thief commits a robbery that night

(2) The thief does not commit a robbery that night

(3) The thief may or may not commit a robbery that night

(4) None of these

Solution:

Let $A$ be the event that the thief sees a policeman. $B$ be the event that he commits a robbery. We have been given that $A \rightarrow ~B$. However, we do not know anything about $~A$, and the question asks us what $B$ will be if $~A$.

∴ We cannot say anything about whether or not the thief will commit a robbery.

Hence, option 1 is the correct choice.

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2.3. IF NOT A, THEN B

This class of statements tells us that an event $B$ will happen if an event $A$ does not happen before it.

Using our example, $~A \rightarrow B$ means that:

‘If it is not raining, then Mukesh is carrying an umbrella’


2.3.1. Representation

This statement is written as $~A \rightarrow B$.

The statement also implies $~B \rightarrow A$.

Thus, if the non-occurrence of an event $A$ implies the occurrence of an event B, then the non-occurrence of event $B$ implies that event $A$ happened.

Let us look at the different ways in which this statement can be interpreted:


First method of approach:

Looking at all possible cases from the point of view of the first event. We are given that $~A \rightarrow B$. The possible cases are:

i. $A$ happens: In this case, $B$ may or may not happen.

In our example, this means that:

If it is raining, then Mukesh may or may not be carrying an umbrella.

ii. $A$ does not happen: In this case, we know directly from the statement that $B$ happens.

In our example, this means that:

If it is not raining, then Mukesh is definitely carrying an umbrella.


Second method of approach:

Looking at all possible cases from the point of view of the second event. We are given that $~A \rightarrow B$, hence we also know that $~B \rightarrow A$.

i. $B$ happens: In this case, $A$ may or may not have happened.

In our example, this means that:

If Mukesh is carrying an umbrella, then it may or may not be raining.

ii. $B$ does not happen: In this case, $A$ must have happened.

In our example, this means that:

If Mukesh is not carrying an umbrella, then it is definitely raining.

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Example:

If Ashok does not wake up early, his father drops him to school by car. One day, Ashok’s father drops him to school by car. Which of the following is true?

(1) Ashok woke up early

(2) Ashok did not wake up early

(3) Ashok may or may not have woken up early

(4) None of these

Solution:

Let $A$ be the event that Ashok wakes up early and $B$ the event that his father drops him by car. We have $~A \rightarrow B$, the negation of which is $~B \rightarrow A$.

We have no expression where $B$ is on the LHS.

So, we cannot say whether or not Ashok has woken up early on a day when his father drops him by car.

Hence, option 3 is the correct choice.

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2.4. IF NOT A, THEN NOT B

This class of statements tells us that if an event $A$ does not happen, event $B$ cannot happen after it.

Using our example, $~A \rightarrow ~B$ means that:

‘If it is not raining, then Mukesh is not carrying an umbrella.’


2.4.1. Representation

This statement is written as $~A \rightarrow ~B$.

This statement also implies that $B \rightarrow A$. Thus, if the non-occurrence of an event $A$ implies non-occurrence of an event $B$, then the occurrence of event $B$ implies that event $A$ happened.

Let us look at the different ways in which this statement can be interpreted:


First method of approach:

looking at all possible cases from the point of view of the first event. We are given that $~A \rightarrow ~B$. The possible cases are:

i. $A$ happens: In this case, $B$ may or may not happen.

In our example, this means that:

If it is raining, then Mukesh may or may not be carrying an umbrella.

ii. $A$ does not happen: In this case, we know that $B$ does not happen.

In our example, this means that:

If it is not raining, then Mukesh is definitely not carrying an umbrella.


Second method of approach:

looking at all possible cases from the point of view of the second event. We are given that $~A \rightarrow ~B$, hence we are also given that $B \rightarrow A$.

i. $B$ happens: In this case, $A$ must have happened.

In our example, this means that:

If Mukesh is carrying an umbrella, then it is definitely raining.

ii. $B$ does not happen: In this case, $A$ may or may not have happened.

In our example, this means that:

If Mukesh is not carrying an umbrella, then it may or may not be raining.

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Example:

Fiza will not wear a hat if it is not sunny. On a particular day, it is sunny. Which of the following statements is true?

(1) Fiza wears a hat

(2) Fiza does not wear a hat

(3) Fiza may or may not wear a hat

(4) None of these

Solution:

Let $A$ be the event that Fiza wears a hat and $B$ the event that it is sunny. Then, we have $~B \rightarrow ~A$

This also implies that $A \rightarrow B$.

Thus, if Fiza wears a hat, it is sunny. However, it does not mean that Fiza wears a hat whenever it is sunny.

∴ If it is sunny, Fiza may or may not wear a hat.

Hence, option 3 is the correct choice.

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3. Selection Problems

In a selection problem, we are given certain objects. We are then given a list of conditions (each of which is a statement like the ones studied in the first part of this chapter) and asked to select a smaller group of objects from among the given objects. Questions are generally asked on the possible composition of the smaller group.

The main challenge in a selection puzzle is to combine the different logical conditions correctly. One must take care to ensure there are no mistakes while translating the statements into logical notations. After that, if the notations are accurately combined, selection problems should not pose difficulty.

Let us look at an example of such a problem.

A group of five boys – Ahmed, Balakrishnan, Chandra, Deven and Eeshwar volunteered for a group activity that requires a group of three members.

The conditions to be taken care of while forming the group are:

  1. If Ahmed is selected, Chandra should be selected and vice-versa.
  2. If Deven is selected, Balakrishnan should not be selected.
  3. If Eeshwar is not selected, Balakrishnan should not be selected.
  4. If Balakrishnan is selected, Ahmed should not be selected.
  5. If Eeshwar is selected, Balakrishnan should be selected

Now, let us find a unique solution to this problem by the following steps:

1. First, identify the comprehensive list of events. For example, if the question deals with selecting people for a team, our events would be a selection of a particular person for the team.

In our example, let the 5 events be:

A: Ahmed is selected

B: Balakrishnan is selected

C: Chandra is selected

D: Deven is selected

E: Eeshwar is selected

2. Represent each separate statement in conditional notation.

In our example, the data statements converted into conditional notation will appear like this:

1. $A \rightarrow C$ and $C \rightarrow A$

2. $D \rightarrow ~B$

3. $~E \rightarrow ~B$

4. $B \rightarrow ~A$

5. $E \rightarrow B$

3. If required, introduce short forms for the names of the different objects.

In our example, we are already using the letters $A$ to $E$ to represent our objects, more appropriately, to represent the event of selection of that particular object.

4. After this, there are two possible methods of solving. In the combination method, we try to combine the logical conditions derived from each statement. We thus convert a large number of simple logical statements to a small number of complex statements. Thus, our information gets organized in a compact form which is helpful in answering questions. This method has the advantage of being short. On the flipside, one must be very careful not to make mistakes in this method, as a significant portion of it involves memorizing facts.

In our example, let us start with the 3rd conditional statement:

$~E\rightarrow ~B$

This means that $B \rightarrow E$

When we combine this with the 4th conditional statement

$B \rightarrow ~A$

We get a group that has $B$, $E$ but not $A$.

Also, since $A \rightarrow C$ and $C \rightarrow A$ (from the 1st conditional statement), the group now contains $B$ and $E$ but not $A$ and $C$.

Now, for the group to be complete, we need 3 members which imply that $D$ would have to be a part of the group.

But, the second conditional statement says that

$D \rightarrow ~B$

This implies that $B \rightarrow ~D$

This means that if $B$ is a part of the group, $D$ cannot be a part of the group.

Thus, the group cannot be formed with $B$, $E$ and $D$.

Also, since it has been inferred that $B \rightarrow E$ and $E \rightarrow B$, it means that both $B$ and $E$ have to be part of the same group or both of them will be excluded from the final group.

Since there is no condition relating $D$ with either $A$ or $C$, we can put the three of them- $A$, $C$ and $D$ in one group.

5. In the table method, we draw a table with three columns. The entries in the first column are the names of all the events and their logical negations. The table tells us which events are ruled out and which events will definitely happen in the event that a certain event does/does not happen. All possible cases are dealt with. This is a rather time-consuming method. However, the chances of errors are significantly lesser.

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Example:

There are nine people numbered from 1 – 9. A group of five people is to be selected out of these nine people. The following conditions apply:

(1) If 2 is selected, 4 and 5 must also be selected

(2) If 3 or 4 or both are selected, 1 must be selected.

(3) If 9 is not selected, 8 will not be selected

(4) If 7 is selected, 1 will not be selected

(5) If 6 is selected, 2 will also be selected


Analysis of the conditions:

We first convert all the statements to logical notation

(1) If 2 is selected, 4 and 5 must also be selected: 2 $\rightarrow$ 4 $\wedge$ 5.

(2) If 3 or 4 or both are selected, 1 must be selected: 3 $\vee$ 4 $\rightarrow$ 1.

(3) If 9 is not selected, 8 will not be selected: ~9 $\rightarrow$ ~8. It also implies 8 $\rightarrow$ 9.

(4) If 7 is selected, 1 will not be selected: 7 $\rightarrow$ ~1. It also implies 1 $\rightarrow$ ~7.

(5) If 6 is selected, 2 will also be selected: 6 $\rightarrow$ 2. It also implies ~2 $\rightarrow$ ~6.

Now, we try the combination approach to the problem.

Combining (1) and (2): (1) tells us that 2 $\rightarrow$ 4 $\wedge$ 5. From (2), we know that 1 will be selected if 3 or 4 or both are selected. Thus, if 2 is selected, 4, 5 and 1 will be selected. This can be written as 2 $\rightarrow$ 4 $\wedge$ 5 $\wedge$ 1. Call this statement (6).

Combining (5) and (6): Since the selection of 6 implies the selection of 2, selection of 6 will also imply the selection of 1, 4 and 5.

This can be written as 6 $\rightarrow$ 1 $\wedge$ 2 $\wedge$ 4 $\wedge$ 5. Call this statement (7).

At this point, we can see that if 6 is a part of the group, the whole group is determined.

Combining (4) and (7): We know that 7 $\rightarrow$ ~1 or 1 $\rightarrow$ ~7, and that 6 $\rightarrow$ 1 $\wedge$ 2 $\wedge$ 4 $\wedge$ 5. Thus, if it is known that 6 is a part of the group, 7 cannot be a part of the group. Further, if 7 is a part of the group, 6 cannot be a part of the group.

Thus, 7 $\rightarrow$ ~6 and 6 $\rightarrow$ ~7. Call this statement (8).

IMPORTANT:

We can now say that, if 3, 7, 8 or 9 are known to be a part of the group, 6 cannot be a part of the group. The reason: If 6 is a part of the group, the members of the group are 1, 2, 4, 5 and 6.

Statement (3) stays as it is. We see that we have derived three new statements (6), (7) and (8) from our original five. These statements offer us information in a more compact and accessible form.

Having analysed all the information available, we move to the questions.


Question 1:

If 6 is selected in the group, which of these will definitely not be a part of the group?

(1) 1       (2) 2       (3) 4       (4) 5       (5) None of these

Solution:

Statement (7) tells us that 6 $\rightarrow$ 1 $\wedge$ 2 $\wedge$ 4 $\wedge$ 5. Thus, all the people in the given options will be a part of the group.

Hence, option 5 is the correct choice.

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Question 2:

If 1, 2, 4 and 5 are four of the members selected in the group, who is the fifth?

(1) 3     (2) 6     (3) 8     (4) 9     (5) Insufficient information

Solution:

We have been given that 6 $\rightarrow$ 1 $\wedge$ 2 $\wedge$ 4 $\wedge$ 5, but not that 1 $\wedge$ 2 $\wedge$ 4 $\wedge$ 5 $\rightarrow$ 6. We can not know for sure who the fifth member of the group could be.

Hence, option 5 is the correct choice.

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Question 3:

If 8 is selected in the group, for how many people (out of 9) can we say for sure whether or not they are selected for the group?

(1) 2     (2) 3      (3) 4     (4) 5     (5) None of these

Solution:

Statement (3) tells us that 8 $\rightarrow$ 9. Thus, we know that 9 will be a part of the group. The note after Statement (8) tells us that 6 cannot be a part of the group if 8 is a part of the group.

Now, if we select 2, 4 and 5 will have to be selected, which also means 1 has to be selected as 2 $\rightarrow$ 4 $\wedge$ 5 $\wedge$ 1. But this will make the number of members in the group 6. Therefore, 2 cannot be selected.

If 7 is selected, 1 cannot be selected. This also means that 4 and 3 will not be selected and consequently 2 will not be selected.

7 $\rightarrow$ ~1 $\rightarrow$ ~4 $\wedge$ ~3 $\rightarrow$ ~2

This means that there will not be enough members in the group if 7 is selected. Therefore, 7 cannot be selected.

The possible combinations now will be 8, 9, 1, 3, 4 or 8, 9, 1, 3, 5 or 8, 9, 1, 4, 5.

Therefore, we can say for sure that 1, 8 and 9 are selected and that 6, 7 and 2 are not selected.

∴ For 6 people out of 9 can we say for sure whether or not they are selected in the group.

Hence, option 5 is the correct choice.

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Question 4:

Which of the following cannot be a part of the same group?

(1) 7, 2      (2) 8, 9     (3) 1, 6      (4) 1, 3     (5) 1, 2

Solution:

We know from statement (4) that 7 $\rightarrow$ ~1.

From statement (2), 3 $\vee$ 4 $\rightarrow$ 1

∴ ~1 $\rightarrow$ ~3 $\wedge$ ~4

Also, from (1), 2 $\rightarrow$ 4 $\wedge$ 5

∴ ~4 $\vee$ ~5 $\rightarrow$ ~2

∴ 7 and 2 cannot be a part of the same group.

Hence, option 1 is the correct choice.

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4. Distribution Problems

A distribution problem combines what we have learnt in the logical selection with the concepts of linear and complex arrangement. We have various objects which we have to distribute among different subgroups.

In a pure Arrangement problem, we were given conditions about the relative placement of different objects. The only difference in a distribution problem is that the conditions require knowledge of logical selection as well.

It is important, therefore, to review the concepts of both logical selection as well as linear and complex rearrangement before starting with logical distribution.

This is a step-by-step procedure for solving distribution problems:

  • Translate all the statements given into logical conditions using notations.
  • Use the method of combination discussed in logical selection to make a transition from many short statements to a few complex statements.
  • Use the above complex statements to write possible cases. Note that the method of writing possible cases is very similar to what we learnt in a linear arrangement. For example, if we are supposed to distribute four people in two houses, with two people living in each house, we can depict the initial case as:
    • House 1: - -
    • House 2: - -
    • Now, if one of our conditions tells us that A lives in house 1, and would not like to live with B, we can fill in the information like this:
    • House 1: A -
    • House 2: B -
  • After taking all our conditions into account, we make linear diagrams for all possible cases and use them to answer the given questions.

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Example:

The Head of a newly formed government desires to appoint five of the six elected members A, B, C, D, E and F to portfolios of Home, Power, Defence, Telecom and Finance. F does not want any portfolio if D gets one of the five. C wants either Home or Finance or no portfolio. B says that if D gets either Power or Telecom then she must get the other one. E insists on a portfolio if A gets one.

Question 1:

Which is a valid assignment?

(1) A-Home, B-Power, C-Defence, D-Telecom, E-Finance

(2) C-Home, D-Power, A-Defence, B-Telecom, E-Finance

(3) A-Home, B-Power, E-Defence, D-Telecom, F-Finance

(4) B-Home, F-Power, E-Defence, C-Telecom, A-Finance

Solution:

The best method here is to eliminate the options on the basis of given information.

Option 1 cannot be the answer as C gets Defence.

Option 3 cannot be the answer as F cannot be with D.

Option 4 cannot be the answer as C gets Telecom.

Option 2 is a valid assignment as it satisfies all the given conditions.

Hence, option 2 is the correct choice.

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Question 2:

If A gets Home and C gets Finance, then which is NOT a valid assignment for Defence and Telecom?

(1) D-Defence, B-Telecom

(2) F-Defence, B-Telecom

(3) B-Defence, E-Telecom

(4) B-Defence, D-Telecom

Solution:

If D gets Telecom then B must get Power.

∴ Option 4 is not a valid assignment.

Hence, option 4 is the correct choice.

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