Logical Reasoning and Constraint Deduction Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Logical Reasoning and Constraint Deduction notes for GATE DA: 19 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Logical Reasoning and Constraint Deduction

Chapter Journey: Logical Reasoning and Constraint Deduction

Step 1: Monotonicity-Based Logical Deduction

Understanding Strict vs. Non-Strict Monotonicity
Reversing Inequalities (Output trends to Input bounds)
The Single Peak and Single Valley Patterns

Step 2: Constraint-Based Sequencing and Elimination

Building Ordering Chains (A is before B, B is after C)
The Elimination Grid Method
Handling "Not" and "Either-Or" Constraints

The Core of Monotonicity

What is Monotonicity?
A function

f (x)

is monotonic if it preserves or reverses the order of its inputs. It never changes direction.

1. Monotonically Increasing

x

gets larger,

f (x)

gets larger (or stays same).

2. Monotonically Decreasing

x

gets larger,

f (x)

gets smaller (or stays same).

The Logical Superpower:
If a function is strictly increasing, it creates a perfect two-way mirror between inputs and outputs.

x_{1} < x_{2} ⟺ f (x_{1}) < f (x_{2})

This means if you are given an inequality about the outputs, you can instantly translate it into an inequality about the inputs.

Strict vs. Non-Strict Monotonicity

Strictly Increasing

Rule:

x_{1} < x_{2} ⟹ f (x_{1}) < f (x_{2})

Consequence: If

f (a) = f (b)

, then

a = b

. (The function never flattens out).

Monotonically Increasing (Non-Strict)

Rule:

x_{1} < x_{2} ⟹ f (x_{1}) \leq f (x_{2})

Consequence: If

f (a) = f (b)

a

and

b

could be different. The function might just be flat (constant) between

a

and

b

Exam Translation:
• "Always grows" / "Strictly increases"

⟹

Strict.
• "Never decreases" / "Monotonically increases"

⟹

Usually Non-Strict (allows flat regions), but context matters. In many competitive exam puzzles, "monotonically increases" implies strict unless a flat region is explicitly hinted at. Always default to strict for inequalities, but keep non-strict in mind for equalities.

The Single Peak and Valley Patterns

Piecewise Monotonicity

Real-world functions (like weight over a lifetime, or profit over time) often increase, reach a maximum, and then decrease.

The Single Peak Rule:
Let

f (x)

strictly increase for

x < P

and strictly decrease for

x > P

.
If

f (a) = f (b)

and

a \neq = b

:
• It is impossible for both

a

and

b

to be

< P

.
• It is impossible for both

a

and

b

to be

> P

.
• Conclusion: One input must be strictly less than

P

, and the other must be strictly greater than

P

The Single Valley Rule:
Let

f (x)

strictly decrease for

x < V

and strictly increase for

x > V

.
If

f (a) = f (b)

and

a \neq = b

:
• Conclusion: One input must be

< V

, and the other must be

> V

Logical Reasoning and Constraint Deduction: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Analytical Aptitude MCQ

A function $h (x)$ monotonically increases up to $x = 10$ and then monotonically decreases. If $h (4) = h (y)$ for some $y \neq = 4$ , which of the following is a possible value for $y$ ?

A.
2
B.
8
C.
12
D.
10

Correct Answer:

Step-by-Step Solution

Key idea: This is a single peak pattern question, recognizable by the function increasing then decreasing, and an equality between outputs at different inputs.

Step 1: Understand the shape of the function. It increases up to $x = 10$ (the peak) and then decreases.

Step 2: Analyze the given condition. We have $h (4) = h (y)$ with $y \neq = 4$ . The input $x = 4$ is on the increasing side of the peak (since $4 < 10$ ).

Step 3: Apply the single peak rule. For a function with a single peak, if two different inputs yield the same output, one input must be on the increasing side (left of the peak) and the other must be on the decreasing side (right of the peak).

Step 4: Since $x = 4$ is on the left, $y$ must be on the right side of the peak. Therefore, $y$ must be strictly greater than 10.

Step 5: Evaluate the options. 2, 8, and 10 are all $\leq 10$ . Only 12 is $> 10$ .

Answer: C

Question 2 · Analytical Aptitude MCQ

Consider a function $W (a)$ representing weight vs age. It strictly increases until age 50 and strictly decreases thereafter. If two brothers have the same weight but different ages, which statement MUST be true about their ages $a_{1}$ and $a_{2}$ (with $a_{1} < a_{2}$ )?

A.
Both are less than 50.
B.
Both are greater than 50.
C.
One is less than 50 and the other is greater than 50.
D.
Both are exactly 50.

Correct Answer:

Step-by-Step Solution

Key idea: Single Peak Pattern (Card c004/c006).

Step 1: The function has a single peak at $a = 50$ .

Step 2: On the left side ( $a < 50$ ), the function is strictly increasing. Thus, every weight value is unique.

Step 3: On the right side ( $a > 50$ ), the function is strictly decreasing. Thus, every weight value is unique.

Step 4: For two different ages to have the same weight, one must be on the increasing slope and the other on the decreasing slope.

Step 5: Therefore, one age is $< 50$ and the other is $> 50$ .

Answer: C

More notes in this unit

Propositional Logic and Logical Equivalence Notes

chapter

Logical Reasoning and Constraint Deduction Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Logical Reasoning and Constraint Deduction notes for GATE DA: 19 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice quest

A question from this chapter

Question 1

A function h (x) monotonically increases up to x = 10 and then monotonically decreases. If h (4) = h (y) for some y \neq = 4, which of the following is a possible value for y ?

Question 2

Consider a function W (a) representing weight vs age. It strictly increases until age 50 and strictly decreases thereafter. If two brothers have the same weight but different ages, which statement MUST be true about their ages a_{1} and a_{2} (with a_{1} < a_{2})?

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