Basics of Logical Reasoning


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    A common notion is that we cannot improve our Logical reasoning skills. Either we know it, or we don’t. We have to live with what we have. Well, we disagree. Every section in an aptitude test has art and science associated with it. Knowing the formulae and concepts is a science while applying the best method is an art. How we score depends on how well we trained the ‘scientist’ and ‘artist’ within us. We are using reasoning principles in our day to day life. We avoid roads during peak traffic hours and we decide to take an umbrella because it is cloudy. We depend on our past experiences to reach conclusions/assumptions which may or may not be true.

    It is very well possible to frame a strategy for tackling logical reasoning questions. Let’s understand some basic elements of reasoning.

    Deductive reasoning: Deductive reasoning is a top-down approach where we reach a logical conclusion from one more general statement (premise). Some interesting deductive reasoning techniques are Law of detachment and Law of syllogism.

    Law of Detachment: There will be a condition and a statement from which a conclusion can be arrived.

    All noble gases are colorless (condition)
    Neon is a noble gas (statement)
    Hydrogen is colorless (statement)
    Neon is colorless (correct conclusion)
    Hydrogen is a noble gas (wrong conclusion)

    Law of syllogism: Combines conditions to reach a conclusion

    If there is a strike, buses wont ply (condition 1)
    If buses wont ply, schools wont function (condition 2)
    If there is a strike, schools wont function (conclusion)

    Valid argument: if the truth of the premises LOGICALLY guarantees the truth of the conclusion. Premises of an argument do not have ACTUALLYto be true in order for the argument to be valid

    A student needs to score more than 90% to pass the test (condition)
    Arun passed the test (statement)
    Arun scored more than 90% (conclusion)

    Here, if condition and statement are true conclusion should be true. Now we don’t know whether a student ACTUALLY needs 90% to pass the test. But the truth of given statements logically guarantees the truth of the conclusion. Hence this is a valid argument.

    Valid arguments generally take the forms,

    If P then Q
    P
    Therefore Q

    If P then Q
    Not Q
    Therefore Not P

    Either P or Q
    Not P
    Therefore Q

    If P then Q
    If Q then R
    Therefore If P then R

    Invalid argument: For an invalid argument. Truth of the statements does not guarantee the truth of conclusion. Invalid arguments take the form

    A is X
    B is X
    Therefore, A is B.

    Odd numbers are integers (statement 1)
    Even numbers are integers (statement 2)
    Odd numbers are even numbers ( Wrong conclusion)

    Sound argument: if and only if an argument is valid and all of its premises are ACTUALLY true.

    Inductive reasoning: Inductive reasoning is a bottom up approach to reach conclusions. Inductive reasoning tries to establish the PROBABLE truth of the conclusion from the premises while deductive reasoning DEFINITIVELY establishes the conclusion based on the premises.

    What we have seen is the ability of these cells to feed the blood vessels of tumors and to heal the blood vessels surrounding wounds (statement)
    these adult stem cells may be an ideal source of cells for clinical therapy (conclusion)

    In the above example there is a sense of a generalized judgment, which may or may not turn out to be true.

    With this understanding we will solve some puzzles. While solving we may not use any of these jargons… we just solve. The level of reasoning in aptitude test can vary from direct questions based on the reasoning laws to complicated stuffs. But most of the questions take a moderate level where we will be given premises from which we can derive further clues to reach the conclusion.

    1) There was a robbery in which a lot of goods were stolen.
    The robber(s) left in a truck.
    It is known that:
    (1) Nobody else could have been involved other than A, B and C.
    (2) C never commits a crime without A's participation.
    (3) B does not know how to drive.

    So, is A innocent or guilty?

    From statement 3,
    B cannot drive. But robbers left in a truck. Some robber should drive.
    From statement 1,
    no one other than A, B, C can be involved. A or C is a robber who drove.
    From statement 2,
    A and C are together, i.e. either both are innocent or both guilty. So A is guilty.

    2) There are twelve billiard balls, all the same size, shape and color. All weigh exactly the same, except that one ball is slightly heavier than others, but not noticeably so in the hand. What is the minimum number of weighing (using a two-pan balance scale) needed to identify the heavy one?

    Statements given:
    12 identical balls except one ball is little heavy.
    A two pan balance is given.

    Conclusion needed:
    Minimum number of weighing to identify the heavy ball.

    Most important point is to identify the characteristics that will help us to reach a conclusion. In this case, it is weight of the balls. We have a two pan weighing balance.  We need to identify the heavy ball and we need to do that in MINIMUM measurements. As we are asked to find the RELATIVE weights we can use BOTH the Pans to see whether one ball is heavier or lighter than the other one by keeping one ball on each pan. As we are asked to use MINIMUM measurements, we need to find the relative weights in terms of GROUPS instead of individual balls. Grouping should be done in such a way that after every measurement we should be able to uniquely identify the group that contains the heavy ball. So that other groups can be eliminated.

    We can have 3 groups with equal number of balls. We can place two groups on the pan (each pan having a group placed on it) and the third one outside. If pan weigh both groups equal we know the heavy ball is in the group outside. Else the heaviest group is the one that contains the heavy ball. Means we can come down to one group, 4 balls.

    We cannot split 4 balls into 3 equal groups; hence we will split it to 2 equal groups. That is 2 balls each. Place one group (of 2 balls each) on each pan. Pick the heavy group, and again split into two equal groups (of one ball each) as we cannot split into three equal groups. Weigh them on the pan with one group on each pan. Heaviest pan in the THIRD measurement decides our ‘Heavy weight Champion’ :-)

    This puzzle is generally referred as ‘12 Billiard balls puzzle’. Different variance of this puzzle is available… try out… and please do share them too :-)

    3) You have 10 bags with 100 coins each. Each coin weighs 1 gram. But in one of the bags all the 100 coins weight 1.1 gram. If you have an electronic weighing machine, what is minimum number of measurements you need to identify the bag with the forgeries?

    We are asked to find the RELATIVE weight of bags (to identify the heaviest bag) in MINIMUM measurements. We have an electronic weighing machine (one ACTUAL measurement at a time). In the previous problem we had to rely totally on the weight of the ball as there was no other unique property which was given or could be introduced. But here there is another attribute for every bag, i.e. number of coins. Now it is not a unique attribute as all bags have same number of coins. But who stops us from making it unique. :-)

    Each coin weigh 1 gram and the heavy one weigh 1.1 gram. if we form a mix of the coin, the extra weight will come from the NUMBER of heavy coins in it. This is our way out  :-)

    Represent each bag with different number of coins (hence making number of coins a unique attribute for each bag). Take 1 coin from Bag1, 2 coins from Bag2, 3 coins from Bag 3… 10 coins from Bag 10. Weigh this mix of coin.

    We weighed 55 coins (1 + 2 + 3 … + 10). If the weight of 55 coins came as 55.7 grams we got 0.7 grams extra. Each heavy coin can add 0.1 gram extra, hence there should be 7 Heavy coins to get 0.7 grams extra weight, means heavy bag is the 7th bag ( as we took 7 coins from 7th bag). Similarly if we get weight as 55.5 heaviest bag is 5th bag and so on…

    4) There are three switches downstairs. Each corresponds to one of the three light bulbs in the attic. You can turn the switches on and off and leave them in any position.  How would you identify which switch corresponds to which light bulb, if you are only allowed one trip upstairs?

    This is a lateral thinking question, which is common during campus placements. These type of question checks our ability to ideate more clues from the given statements. Think outside box!

    What are the attributes of a bulb that can be controlled by a switch? Its state (ON/OFF). But we need to deal with THREE bulbs in one go. Hence we cannot go with just ON/OFF. We need to go deep to the available attributes and see whether there are any other attribute that get affected by being ON and OFF. Temperature of the glass… more time a bulb is ON more the temperature of its glass will be.  

    we can solve this problem as,

    Keep the first switch OFF.
    Keep second switch ON for some time.
    Turn OFF the second switch, immediately Turn ON the third switch and RUN to attic.

    Now the bulb which is ON corresponds to third switch. Bulb which is OFF and HOT corresponds to second switch and bulb which is OFF and NOT HOT corresponds to the first switch.

    5) Persons X, Y, Z and Q live in red, green, yellow or blue colored houses placed in a sequence on a street. Z lives in a yellow house. The green house is adjacent to the blue house. X does not live adjacent to Z. The yellow house is in between the green and red houses. The color of the house X lives in is:

    a. Blue

    b. Green

    c. Red

    d. Not possible to determine   (CAT 2000)

    Statements given:

    Statement1: four houses colored Red, Green, Yellow and Blue placed on a sequence

    Statement2: Z lives in Yellow house.
    Statement3: Green is Adjacent to Blue.
    Statement4: X does not live adjacent to Z
    Statement5: Yellow is between Green and Red.

    What we need? Where X lives

    Always scribble down the given statements into some easily readable/referable format. Going back to re-read the statements will waste our precious time.

    Statement 1: Four houses colored Red, Green, Yellow and Blue placed on a sequence. Just assume some pattern to start… we will tweak it as we go.

    Statement 2: Z lives in Yellow house.

      

    Statement 3: Green is Adjacent to Blue.

    Now this opens multiple possibilities. We will use it later once we have narrowed the situation.

    Statement 5: Yellow is between Green and Red.

    We have two possibilities

    Now we can use statement 3, and eliminate the second possibility. Remaining one is

    Statement 4: X does not live adjacent to Z

    X cannot live in RED or GREEN… hence X can only live @ Blue house.

     

    Actually we can solve this one just by inspection without drawing any figures. We need to connect X with a House. Statement 4 connects X with Z (another person) then statement 2 connects Z with a house (Yellow). We know X is not in yellow. Statement 3 connects Yellow with 2 other houses (Green and Red) and statement 4 makes it impossible for X to be in Green and Red and only option remaining is Blue.  Still I thought little explanation may be helpful to understand the thought process behind such questions. :-)

    Logical reasoning is Fun. But ensure we don’t waste time in the thrill of connecting the dots. Regular practice helps a lot to fine tune our thought process in logical reasoning…

    Happy Learning :-)


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