Logical Reasoning for GATE

Logical Reasoning

Logical reasoning tests how well you think step-by-step. Like solving a puzzle, you follow clues to reach a conclusion.

---

Syllogisms — Are the Conclusions Valid?

What is a syllogism? Two statements (premises) → Is a conclusion valid?

The Venn Diagram Method — Always Draw This!

Statement: "All cats are animals"
Draw: Cats circle INSIDE Animals circle
       _______________
      |   Animals     |
      |  _________    |
      | |  Cats   |   |
      | |_________|   |
      |_______________|

Statement: "Some animals are wild"
Draw: Animals and Wild circles OVERLAPPING
       _______   _______
      |       | |       |
      |Animals|○| Wild  |
      |       | |       |
      |_______| |_______|
      (○ = overlapping region = "some")

The 4 Statement Types: | Statement | Venn diagram | |-----------|-------------| | All A are B | A circle is entirely inside B | | Some A are B | A and B overlap (partially) | | No A are B | A and B are completely separate | | Some A are not B | Part of A is outside B |

The 3 Key Rules: 1. If both premises have "some" → no definite conclusion possible 2. At least one premise must be positive for a positive conclusion 3. At least one premise must be universal ("All"/"No") for a definite conclusion

Worked Example:

Premise 1: All doctors are educated.
Premise 2: Some educated people are rich.
Conclusion: Some doctors are rich — VALID?

Draw: Doctors ⊂ Educated; Educated ∩ Rich = some overlap
The overlap might include the doctors region or might not.
→ Conclusion is NOT definite — answer: "may or may not be true"

---

Blood Relations — Draw the Family Tree!

The only strategy: Draw as you read. Never solve mentally.

Family tree symbols:
   ■ = Male    ● = Female    ■-● = Married couple
   |           |
   ↓           ↓  = children

Common relationships: | Relationship | Meaning | |-------------|---------| | Sibling | Same parents | | Cousin | Your parent's sibling's child | | Uncle/Aunt | Your parent's sibling | | Nephew/Niece | Your sibling's son/daughter | | In-laws | Spouse's family |

Example: "A is B's father. C is A's sister. D is C's daughter. What is D to B?"

Draw: A(male) — sister — C(female)
      |                      |
      B                      D

A is father of B.
C is A's sister = B's aunt.
D is C's daughter = B's cousin.
Answer: Cousin

---

Directions — Fix Your Compass First!

The one rule: Always draw a compass rose and track the person's facing direction.

        North (N)
           ↑
West(W) ←  +  → East(E)
           ↓
        South (S)

Turning:

Facing North + turn Right (clockwise) = now facing East
Facing North + turn Left (anticlockwise) = now facing West

Shadows:

Morning → Sun is in EAST → shadow falls WEST
Evening → Sun is in WEST → shadow falls EAST
Noon    → Sun overhead → shadow falls SOUTH (Northern hemisphere)

---

Number Series — The Pattern Hunt

Step 1: Look at differences between consecutive terms. Step 2: If differences don't form a pattern, look at ratios. Step 3: Check if it's TWO interleaved series.

Pattern	Example	Next term
Constant difference	3, 7, 11, 15, ___	19 (+4 each time)
Geometric (×ratio)	2, 6, 18, 54, ___	162 (×3 each time)
Differences form series	1, 2, 4, 7, 11, ___	16 (diff: 1,2,3,4,5)
Squares	1, 4, 9, 16, 25, ___	36
Two interleaved	2, 3, 4, 6, 8, 12, ___	16 (two ×2 series)
Fibonacci	1, 1, 2, 3, 5, 8, ___	13 (each = sum of prev 2)

---

Coding-Decoding — Secret Messages

Type 1: Letter Shift

If CAT → FDW (each letter shifted +3)
C+3=F, A+3=D, T+3=W
To decode FDW → subtract 3 → CAT

Type 2: Reverse Alphabet (A↔Z)

A(1) ↔ Z(26)    B(2) ↔ Y(25)    C(3) ↔ X(24)
Formula: Code = 27 - position
A=26, B=25, C=24... Z=1

Type 3: Position Numbers

A=1, B=2, C=3... Z=26
CAT = 3, 1, 20

---

Clocks — Hands as Racers on a Track

Analogy: Imagine the clock face as a circular track. The minute hand is a fast runner, the hour hand is a slow walker. They start together at 12:00.

Minute hand speed: 360°/60min = 6° per minute  ← fast runner
Hour hand speed:   360°/720min = 0.5° per minute  ← slow walker
Relative speed:    6 - 0.5 = 5.5° per minute

Angle between hands at H hours M minutes:

Angle = |30H - 5.5M|  degrees
If > 180°, take 360° - angle (we want the smaller angle)

Example: Find angle at 3:40
= |30×3 - 5.5×40| = |90 - 220| = 130°

When do hands coincide?

Every 65 5/11 minutes (≈ 65.45 minutes)
22 times in 24 hours (not 24 times — they skip once near 12!)

When are hands at 90° (right angle)?

44 times in 24 hours (not 48 — they skip twice near 3 and 9)

---

Calendars — The Odd Days Trick

Odd day = remainder when days are divided by 7.

Ordinary year = 365 days = 52 weeks + 1 day → 1 odd day
Leap year     = 366 days = 52 weeks + 2 days → 2 odd days

Leap year rule:

Year divisible by 4 → leap year
EXCEPT century years → must be divisible by 400
2000 ✓ (÷400), 1900 ✗ (÷400 fails), 2100 ✗, 2400 ✓

Reference point to memorise: January 1, 2000 = Saturday

Day codes: Sun=0, Mon=1, Tue=2, Wed=3, Thu=4, Fri=5, Sat=6

---

Seating Arrangements — The Constraint Approach

Linear arrangement: 1. Fix the most-constrained person (has most conditions) 2. Place others relative to them 3. List remaining positions

Circular arrangement:

n people in a circle = (n-1)! arrangements
(Fix one person — they have no "position" since circle has no start)

Note: "Facing centre/outside" changes the meaning of left/right!

---

Quick Check Questions

Q1. At 4:20, what is the angle between clock hands?

Angle = |30×4 - 5.5×20| = |120 - 110| = 10°

Q2. Series: 2, 5, 10, 17, 26, ___

Differences: 3, 5, 7, 9, 11 (odd numbers)
Next difference = 11, so next term = 26 + 11 = 37

Q3. A walks 3 km North, then 4 km East. How far from start?

Draw right triangle: 3 km vertical, 4 km horizontal
Distance = √(3² + 4²) = √(9+16) = √25 = 5 km