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

Logical Reasoning and Constraint Deduction short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: Monotonicity Deduction Cheat Sheet

1. The Core Rules

Strictly Increasing: $f (a) < f (b) ⟺ a < b$
Strictly Decreasing: $f (a) < f (b) ⟺ a > b$
Strict Equality: If $f$ is strictly monotonic and $f (a) = f (b)$ , then $a = b$ .

2. The Single Peak Rule

Peak at $P$ .
If $f (a) = f (b)$ and $a \neq = b ⟹$ One is $< P$ , the other is $> P$ .

3. The Golden Constraints

No Symmetry:

f (a) = f (b)

does NOT mean the peak is at

\frac{a + b}{2}

No Linearity: Do not assume equal rates of change unless stated.

Translate First: Convert "always grows" to

x < y ⟹ f (x) < f (y)

immediately.

Final Exam Readiness Checklist

Grid First: Never solve in your head. Draw the matrix.

Restrictive Words: "Only", "Exactly", "None" apply globally.

Sequence Type: Is it fixed (seating) or rolling (tournament)?

Order of Operations: Direct/Positive clues

\to

Indirect/Negative clues.

Branching: Stuck on an "OR" condition? Draw two mini-grids.

Verify: Once solved, read every single clue again to ensure no rule was accidentally broken.

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 short notes in this unit

Propositional Logic and Logical Equivalence Short Notes

chapter

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

Logical Reasoning and Constraint Deduction short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice

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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