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Logical Reasoning and Constraint Deduction Short Notes for GATE DA

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

logical reasoning and constraint deduction short notes

Summary: Monotonicity Deduction Cheat Sheet

.summary-card { padding: 1.5rem; background: var(--color-bg); color: var(--color-text); display: flex; flex-direction: column; gap: 1.5rem; } .summary-section { border-bottom: 1px solid var(--color-border); padding-bottom: 1rem; } .summary-section:last-child { border-bottom: none; } .section-title { color: var(--color-primary); font-weight: 700; margin-bottom: 0.8rem; font-size: 1.1rem; } .rule-list { list-style: none; padding: 0; margin: 0; } .rule-list li { margin-bottom: 0.5rem; padding-left: 1.2rem; position: relative; font-family: monospace; font-size: 0.95rem; } .rule-list li::before { content: '>'; position: absolute; left: 0; color: var(--color-primary); font-weight: bold; } .constraint-box { background: color-mix(in srgb, var(--color-muted) 8%, transparent); padding: 0.6rem 0.8rem; border-radius: 4px; margin-bottom: 0.5rem; font-size: 0.9rem; } .constraint-box strong { color: var(--color-primary); } 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

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.

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Question 1

Level 1: Warm-up

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

Level 1: Warm-up

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})?

Question 3

Level 1: Warm-up

Temperature T (h) as a function of altitude h strictly increases up to h = 10 km and strictly decreases thereafter. If two different altitudes h_{1} and h_{2} (h_{1} < h_{2}) have the same temperature, what must be true?

Question 4

Level 1: Warm-up

Let f (x) be a strictly increasing function. If f (a) < f (b), which of the following must be true?

Question 5

Level 1: Warm-up

The function f (x) is strictly decreasing for all real x . If f (2 x - 1) > f (5), which of the following must be true?

Question 6

Level 1: Warm-up

A function g (x) monotonically decreases for x \leq 5 and monotonically increases for x > 5 . If g (2) = g (k) and k \neq = 2, which of the following must be true?

Question 7

Level 1: Warm-up

A function f (x) is strictly decreasing on the interval [- 10, 10] . If f (a) < f (b) for some a, b \in [- 10, 10], which of the following must be true?

Question 8

Level 1: Warm-up

Let f (x) be a strictly increasing function and g (x) be a strictly decreasing function. What is the monotonicity of the composite function h (x) = f (g (x)) ?

Question 9

Level 1: Warm-up

A function f (x) has a single peak. We know that f (2) = f (8) . A student concludes that the peak must be at x = \frac{2 + 8}{2} = 5 . Is this conclusion necessarily correct?

Question 10

Level 1: Warm-up

Let f (x) be a strictly monotonic function. If f (a) = f (b), which of the following must be true?

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Logical Reasoning and Constraint Deduction Short Notes for GATE DA

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 (10 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

Question 3 · Analytical Aptitude MCQ

Temperature $T (h)$ as a function of altitude $h$ strictly increases up to $h = 10$ km and strictly decreases thereafter. If two different altitudes $h_{1}$ and $h_{2}$ ( $h_{1} < h_{2}$ ) have the same temperature, what must be true?

A.
$h_{1} < 10 < h_{2}$
B.
$h_{1} < h_{2} < 10$
C.
$10 < h_{1} < h_{2}$
D.
$h_{1} = 10$ or $h_{2} = 10$

Correct Answer:

Step-by-Step Solution

Key idea: This is a Single Peak Pattern question (Card c006), recognizable by a function that increases then decreases.

Step 1: The function $T (h)$ has a single peak at $h = 10$ .

Step 2: For $h < 10$ , the function is strictly increasing. Thus, every temperature value is unique on this side.

Step 3: For $h > 10$ , the function is strictly decreasing. Thus, every temperature value is unique on this side.

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

Step 5: Since $h_{1} < h_{2}$ , it must be that $h_{1} < 10$ and $h_{2} > 10$ .

Step 6: This matches the condition $h_{1} < 10 < h_{2}$ .

Answer: A

Question 4 · Analytical Aptitude MCQ

Let $f (x)$ be a strictly increasing function. If $f (a) < f (b)$ , which of the following must be true?

A.
$a > b$
B.
$a < b$
C.
$a = b$
D.
Cannot be determined

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct definition question on strictly increasing functions, recognizable by the phrase "strictly increasing" and an inequality between outputs.

Step 1: Recall the definition of a strictly increasing function. As the input $x$ increases, the output $f (x)$ strictly increases.

Step 2: This means the order of the inputs is perfectly preserved in the outputs. If $x_{1} < x_{2}$ , then $f (x_{1}) < f (x_{2})$ .

Step 3: We are given $f (a) < f (b)$ . Because the function is strictly increasing, the inequality between the outputs directly reflects the inequality between the inputs.

Step 4: Therefore, it must be true that $a < b$ .

Answer: B

Question 5 · Analytical Aptitude MCQ

The function $f (x)$ is strictly decreasing for all real $x$ . If $f (2 x - 1) > f (5)$ , which of the following must be true?

A.
$2 x - 1 < 5$
B.
$2 x - 1 > 5$
C.
$2 x - 1 = 5$
D.
Cannot be determined

Correct Answer:

Step-by-Step Solution

Key idea: This is an inequality reversal question, recognizable by the phrase "strictly decreasing" and an inequality between outputs involving algebraic expressions.

Step 1: Recall the property of a strictly decreasing function. As the input increases, the output strictly decreases.

Step 2: This means the order of the inputs is reversed in the outputs. If $A > B$ , then $f (A) < f (B)$ .

Step 3: We are given $f (2 x - 1) > f (5)$ . Because the function is strictly decreasing, the inequality between the outputs is the reverse of the inequality between the inputs.

Step 4: Therefore, the input corresponding to the larger output must be the smaller input. This gives $2 x - 1 < 5$ .

Answer: A

Question 6 · Analytical Aptitude MCQ

A function $g (x)$ monotonically decreases for $x \leq 5$ and monotonically increases for $x > 5$ . If $g (2) = g (k)$ and $k \neq = 2$ , which of the following must be true?

A.
$k > 5$
B.
$k < 5$
C.
$k = 5$
D.
$k < 2$

Correct Answer:

Step-by-Step Solution

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

Step 1: Understand the shape of the function. It decreases up to $x = 5$ (the valley) and then increases.

Step 2: Analyze the given condition. We have $g (2) = g (k)$ with $k \neq = 2$ . The input $x = 2$ is on the decreasing side of the valley (since $2 < 5$ ).

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

Step 4: Since $x = 2$ is on the left, $k$ must be on the right side of the valley. Therefore, $k$ must be strictly greater than 5.

Answer: A

Question 7 · Analytical Aptitude MCQ

A function $f (x)$ is strictly decreasing on the interval $[- 10, 10]$ . If $f (a) < f (b)$ for some $a, b \in [- 10, 10]$ , which of the following must be true?

A.
$a < b$
B.
$a = b$
C.
$a \leq b$
D.
$a > b$

Correct Answer:

Step-by-Step Solution

Key idea: This is an inequality reversal question, recognizable by the phrase "strictly decreasing" and an inequality between outputs.

Step 1: Recall the property of a strictly decreasing function. As the input increases, the output strictly decreases.

Step 2: This means the order of the inputs is reversed in the outputs. If $A > B$ , then $f (A) < f (B)$ .

Step 3: We are given $f (a) < f (b)$ . Because the function is strictly decreasing, the inequality between the outputs is the reverse of the inequality between the inputs.

Step 4: Therefore, the input corresponding to the smaller output must be the larger input. This gives $a > b$ .

Answer: D

Question 8 · Analytical Aptitude MCQ

Let $f (x)$ be a strictly increasing function and $g (x)$ be a strictly decreasing function. What is the monotonicity of the composite function $h (x) = f (g (x))$ ?

A.
Strictly decreasing
B.
Strictly increasing
C.
Constant
D.
Cannot be determined

Correct Answer:

Step-by-Step Solution

Key idea: This is a composition of monotonic functions question, recognizable by the combination of two functions with known monotonicities.

Step 1: Understand the behavior of $g (x)$ . It is strictly decreasing. As $x$ increases, $g (x)$ decreases.

Step 2: Understand the behavior of $f (x)$ . It is strictly increasing. As its input decreases, its output decreases.

Step 3: Trace the composite function $h (x) = f (g (x))$ . As $x$ increases, the inner function $g (x)$ decreases.

Step 4: Because $g (x)$ decreases, the input to the outer function $f$ is decreasing.

Step 5: Since $f$ is strictly increasing, a decreasing input causes a decreasing output.

Step 6: Therefore, as $x$ increases, $h (x)$ decreases. The composite function is strictly decreasing.

Answer: A

Question 9 · Analytical Aptitude MCQ

A function $f (x)$ has a single peak. We know that $f (2) = f (8)$ . A student concludes that the peak must be at $x = \frac{2 + 8}{2} = 5$ . Is this conclusion necessarily correct?

A.
Yes, because all peaks are symmetric.
B.
Yes, because the average of inputs gives the peak.
C.
No, because monotonicity does not imply symmetry.
D.
No, because the function must be linear.

Correct Answer:

Step-by-Step Solution

Key idea: Trap - Assuming Symmetry (Card c007).

Step 1: Recall the property of single-peak functions.

Step 2: We know one input is on the left and one is on the right.

Step 3: However, the rate of increase and decrease can be different.

Step 4: The function might rise steeply and fall gently, or vice versa.

Step 5: Therefore, the peak is not necessarily at the midpoint. Symmetry is not guaranteed by monotonicity.

Answer: C

Question 10 · Analytical Aptitude MCQ

Let $f (x)$ be a strictly monotonic function. If $f (a) = f (b)$ , which of the following must be true?

A.
$a

eq b$
B.
$a = b$
C.
$a > b$
D.
$a < b$

Correct Answer:

Step-by-Step Solution

Key idea: Injective Property of Strict Monotonicity (Card c002).

Step 1: Definition of Strictly Monotonic: It is either strictly increasing OR strictly decreasing everywhere.

Step 2: Both strictly increasing and strictly decreasing functions are one-to-one (injective).

Step 3: This means distinct inputs always produce distinct outputs.

Step 4: Conversely, if outputs are equal ( $f (a) = f (b)$ ), the inputs MUST be equal.

Step 5: Therefore, $a = b$ .

Answer: B

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