Logical Reasoning and Constraint Deduction Notes for GATE DA
Logical Reasoning and Constraint Deduction notes for GATE DA: 17 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice quest
logical reasoning and constraint deduction notes
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
As x gets larger, f(x) gets larger (or stays same).
2. Monotonically Decreasing
As 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. x1<x2⟺f(x1)<f(x2)
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:x1<x2⟹f(x1)<f(x2) Consequence: If f(a)=f(b), then a=b. (The function never flattens out).
Monotonically Increasing (Non-Strict)
Rule:x1<x2⟹f(x1)≤f(x2) Consequence: If f(a)=f(b), a and bcould 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.
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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=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 a1 and a2 (with a1<a2)?
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 h1 and h2 (h1<h2) 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(2x−1)>f(5), which of the following must be true?
Question 6
Level 1: Warm-up
A function g(x) monotonically decreases for x≤5 and monotonically increases for x>5. If g(2)=g(k) and k=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∈[−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=22+8=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 Notes for GATE DA
Logical Reasoning and Constraint Deduction notes for GATE DA: 17 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
As x gets larger, f(x) gets larger (or stays same).
2. Monotonically Decreasing
As 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. x1<x2⟺f(x1)<f(x2)
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:x1<x2⟹f(x1)<f(x2) Consequence: If f(a)=f(b), then a=b. (The function never flattens out).
Monotonically Increasing (Non-Strict)
Rule:x1<x2⟹f(x1)≤f(x2) Consequence: If f(a)=f(b), a and bcould 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=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=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 (10 Problems)
Question 1 · Analytical AptitudeMCQ
A function h(x) monotonically increases up to x=10 and then monotonically decreases. If h(4)=h(y) for some y=4, which of the following is a possible value for y?
A.
2
B.
8
C.
12
D.
10
Correct Answer:
C
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=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 ≤10. Only 12 is >10.
Answer: C
Question 2 · Analytical AptitudeMCQ
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 a1 and a2 (with a1<a2)?
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:
C
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 AptitudeMCQ
Temperature T(h) as a function of altitude h strictly increases up to h=10 km and strictly decreases thereafter. If two different altitudes h1 and h2 (h1<h2) have the same temperature, what must be true?
A.
h1<10<h2
B.
h1<h2<10
C.
10<h1<h2
D.
h1=10 or h2=10
Correct Answer:
A
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 h1<h2, it must be that h1<10 and h2>10.
Step 6: This matches the condition h1<10<h2.
Answer: A
Question 4 · Analytical AptitudeMCQ
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:
B
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 x1<x2, then f(x1)<f(x2).
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 AptitudeMCQ
The function f(x) is strictly decreasing for all real x. If f(2x−1)>f(5), which of the following must be true?
A.
2x−1<5
B.
2x−1>5
C.
2x−1=5
D.
Cannot be determined
Correct Answer:
A
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(2x−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 2x−1<5.
Answer: A
Question 6 · Analytical AptitudeMCQ
A function g(x) monotonically decreases for x≤5 and monotonically increases for x>5. If g(2)=g(k) and k=2, which of the following must be true?
A.
k>5
B.
k<5
C.
k=5
D.
k<2
Correct Answer:
A
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=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 AptitudeMCQ
A function f(x) is strictly decreasing on the interval [−10,10]. If f(a)<f(b) for some a,b∈[−10,10], which of the following must be true?
A.
a<b
B.
a=b
C.
a≤b
D.
a>b
Correct Answer:
D
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 AptitudeMCQ
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:
A
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 AptitudeMCQ
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=22+8=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:
C
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 AptitudeMCQ
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:
B
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.