Coded Inequality for Competitive Exams.
Quick Refresher of the symbols and English Equivalent word of the same.
< –> Less than
≤ –> Less than or Equal to
> –> Greater than
≥ –> Greater than or Equal to
= –> Equal to
Let's learn the topic through simple examples.
Example : 1
Consider the below expression, let's try to establish relationship between any two variables randomly.
A < D > C ≤ E ≥ B
i) Relationship between A and D is : A < D
ii) Relationship between A and C is : Relationship cannot be established. Why?
Note: (1)Thumb Rule is whenever opposite symbols(<,> or ≤,≥) lies between the variables, RELATIONSHIP CANNOT BE ESTABLISHED.
iii) Relationship between C and B is : Relationship cannot be established
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established
iv) Relationship between A and B is : Relationship cannot be established
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
Example : 2
A ≤ D < C < E > F
i)
Relationship between A and D is : A ≤ D.
ii) Relationship between A and C is : A < C. How?
Note: (2)Thumb Rule is whenever "≤" competes with "<" winner is always "<" provided there is no opposite symbol comes in between them.
Same goes for ≥
(2)Thumb Rule is whenever "≥" competes with ">" winner is always ">" provided there is no opposite symbol comes in between them.
iii) Relationship between A and E is : A < E.
iii) Relationship between A and F is : Relationship cannot be established.
Opposite symbols comes in place, so between A and F relationship cannot be established as per Thumb Rule (1).
Example : 3
A ≤ D = C < E ≤ F ≥G
i)
Relationship between A and D is : A ≤ D.
ii) Relationship between A and C is : A ≤ C. How?
Note: (3)Thumb Rule is = symbol just equates one to other. i.e. D = C (or) C = D.
We can ignore if it comes in the middle.
iii) Relationship between A and E is : A<E. Ignore = symbol in these cases.
As per Thumb Rule 2, ≤ competes with < winner is <.
iv) Relationship between A and F is : A<F.
As per Thumb Rule 2, ≤ competes with < winner is <.
v) Relationship between F and C is : F>C.
As per Thumb Rule 2, ≥ competes with > winner is >.
vi) Relationship between F and A is : F>A. (Read from Right to Left in this case)
As per Thumb Rule 2, ≥ competes with > winner is >.
vii) Relationship between C and G is : Relationship cannot be established
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established
Let us solve sample questions being asked in exam(IBPS 2014 PO)
1) Statement : S > M ≥ D > H ≤ R ≤ T < W
Conclusions:
i) S > H
ii) W > H
iii)R < W
iv) M > T
Solution:
i) S > H : True (S > H)
As per Thumb Rule 2, ≥ competes with > winner is >.
Note: No Opposite symbols in between S and H.
ii) W > H : True (W > H)
Note: (Read from Right to Left in this case)
As per Thumb Rule 2, ≥ competes with > winner is >.
iii) R < W : True(R < W)
As per Thumb Rule 2, ≤ competes with < winner is <.
iv) M > T : False (Relationship cannot be established)
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established
Ans is i),ii) & iii) are true.
2) Statement : M > U > L ≤ N ;
L ≥ Y > A
Note: Above two statements are given, there should be always a common variable to relate them. In our case that common variable is L.
We can rewrite the above statement as below:
M > U > L ≤ N
L ≥ Y > A
If they ask you to derive the relation between M and A:
We should traverse like M->U->L->Y->A
Conclusions:
i) Y < N
ii) M > N
iii)N = Y
iv) M > A
Solution:
i) Y < N : False.-> (Y ≤ N) How?
We should traverse from Y->L(bottom)->L(top)->N
Now with the symbols Y ≤ L ≤ N.
From the above statement we can easily check for our case i.e Y < N
Note: Don't interpret Y ≤ N as Y < N and Y = N. Your answer should be definitely true.
So Y ≤ N cannot be true. Since there are two possibility exist, answer is false.
ii) M>N : False (Relationship cannot be established)
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
iii) N = Y : False -> (N ≥ Y)
Note: As i said before, don't interpret N ≥ Y as N > Y and N = Y.
Since there are two possibility exist, answer is false.
iv) M > A : True (M > A)
We should traverse like M >U>L≥Y>A
As per Thumb Rule 2, ≥ competes with > winner is >.
Note: No Opposite symbols in between M and A.
Before choosing the final answer, there is one hidden special case exist in this problem which is nothing but EITHER OR.
Note: (Thumb Rule - 4 )
=>Whenever same variables combo repeats in conclusion multiple times and
if you able to establish the relationship from the given statement for those variables and
if that established relationship satisfies both the conclusions with the OR operator
then answer for those variables is EITHER OR.
else stick with the answer which we calculated individually.
Let's go back to conclusion statements again
i) Y < N and
iii) N = Y can also be written as Y = N
Variables Y and N appears in two options [ i and iii ] with different symbols.
This is the key (or clue) for EITHER OR.
Now from the given statement, relationship between
common variables N and Y is
(N ≥ Y)
N ≥ Y => N > Y (or) N = Y.
Note: Whenever same variables combo(Y and N in this case) repeats in conclusion multiple times(repeated in i) and iii) option) and
if you able to establish the relationship from the given statement(for variable N and Y we found relation as
N ≥ Y from the given statement in this case) and
that established relationship(
N ≥ Y => N > Y (or) N = Y) satisfies both the conclusions with the OR operator then answer for those variables is EITHER OR(in our case either i) or iii) is true).
(Thumb Rule - 4 )
Ans is only iv) and either i) or iii) is true is True.
Few quick example for EITHER OR case
Common variable falls under EITHER OR case
1)Statement: H
≤ L = M
Conclusion :
i) H < M
ii) H = M
From the given statement relationship between H and M is => H ≤ M
If we derive conclusion for i) and ii) individually as we do in general, both will be equated to false.
As i said before clue for EITHER OR case is, if same variables combo repeat in conclusion
multiple times and if both the conclusions are satisfied by the established relationship with the OR operator (H ≤ M => H < M (or) H = M ) then answer is EITHER OR.
Ans : either i) or ii) is True.
Common variable not falls under EITHER OR case
2)Statement: H
≤ L = M > T
Conclusion :
i) H < T
ii) H = T
From the given statement relationship between common variables H and M is
=> Relationship cannot be established
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
In this case we should stick with the answer as we do calculate individually.
i) H < T : False.
Relationship cannot be established
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
ii) H = T : False.
Relationship cannot be established
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
Ans: None is true.
3)
Statement : P > Q ≤ C ≤ B = M > D
Conclusions:
i) M > Q
ii) D < P
iii) M = Q
iv) C > D
Let's quickly find EITHER OR falls in this problem.
Option i) and iii) has common variables (i.e is M and Q).
From the statement, relationship between M and Q is M ≥ Q = > M > Q (or) M = Q
So for i and iii answer is either i or iii is true.
ii) D < P : False.Relationship cannot be established
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established
iv) C > D: False.Relationship cannot be established
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established
Ans: Only either i or iii is true.
4) Statement : J ≥ A > D = L;
L < A < M
Conclusions:
i) M < J
ii) J > L
iii)D > L
iv) A < M
Solution:
i) M < J : False (Relationship cannot be established)
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
ii) J > L : True:
As per Thumb Rule 2, ≥ competes with > winner is >.
Note: No Opposite symbols in between J and L. Only = symbol exist.
iii)D > L : False ( D = L )
From the statement, it is clear D = L.
iv) A < M : False (Relationship cannot be established)
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
Ans is only ii) is True.
5) Statement : M > H ≤ Y ≤ R < U = Z ≥ E.
Conclusions:
i) M < R
ii) Z ≤ R
iii)R > E
iv) Z > H
Solution:No Common variables in multiple conclusions.
i) M < R: False (Relationship cannot be established)
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
ii) Z ≤ R: False (Z > R). From the statement it is clear that Z > R.
iii)R > E : False (Relationship cannot be established)
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
iv) Z > H : True (Z > H)
As per Thumb Rule 2, ≥ competes with > winner is >
Ans : only iv) is true.
6) Statement : Y > F ≤ O ≤ P
F ≥ U < T
Conclusions:
i) Y > P
ii) T < F
iii)O > T
iv) P < U
Solution:
i) Y > P : False (Relationship cannot be established)
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
ii) T < F :False (Relationship cannot be established)
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
iii)O > T : False. We should traverse like O ≥ F ≥ U < T.
As per Thumb Rule(1) opposite symbol comes in place so relationship cannot be established.
iv) P < U : False. From the statement it is clear P ≥ U.
Ans : None is true
By practising few problems, you will get clear idea of how those 4 thumb rules has been arrived. You can solve any problem in less than 3 minute in Exam for sure.
Please post back any queries if you have.
All the Best for your Exams.
Leave Nothing for tomorrow which can be done today.
--Abraham Lincoln.
