Negation Rules

In many AR questions, Set A and Set B each have their own independent rule. But in some questions, Set B is simply defined as "everything that is not Set A." Understanding this distinction — and knowing how to exploit it — can save you significant time on test day.

---

Two Types of Set Relationships

Independent Rules

Set A has its own rule. Set B has a completely separate, unrelated rule. Neither set is defined in terms of the other.

Example:

• Set A: Every box contains a circle
• Set B: Every box has exactly 5 shapes

A test shape could satisfy both rules (contains a circle AND has 5 shapes), one rule, or neither. The "Neither" option exists precisely for cases like this.

Negation Rules

Set B is defined as "everything that does not satisfy Set A's rule" (or vice versa). There is only one rule, and the two sets are complementary.

Example:

• Set A: Total sides is even
• Set B: Total sides is odd

Every possible test shape has either an even or odd total number of sides. There is no "Neither" — but remember, the "Neither" option still exists in the question format. In negation-rule questions, "Neither" is never the correct answer (every test shape fits one set or the other).

---

How to Recognise Negation Rules

Indicator	Meaning
Set A and Set B rules appear to be exact opposites	Likely a negation rule
You cannot construct a test shape that fits neither set	Strongly suggests negation
The rules are complementary (cover all possibilities)	Negation confirmed
"Neither" would logically be impossible	Negation rule

Common Negation Rule Pairs

Set A	Set B
Even number of shapes	Odd number of shapes
Total sides is even	Total sides is odd
All shapes are curved	At least one straight shape
All shapes are the same colour	At least two different colours
Contains a triangle	Does not contain a triangle
All shapes are large	At least one shape is not large
Shapes are symmetrically arranged	Shapes are asymmetrically arranged

Notice the pattern: One set uses "all/every/always" and the other uses "not all/at least one exception/sometimes not." This is the logical structure of negation: the negation of "all X are Y" is "at least one X is not Y."

---

Strategic Advantage of Negation Rules

When you identify a negation rule, you gain a significant speed advantage:

1. You only need to find ONE rule — the second set's rule is automatically the opposite
2. "Neither" is eliminated — you only need to determine Set A or Set B
3. Decision-making is binary — faster than three-way (A / B / Neither)

Speed Protocol for Negation Rules

1. Identify Set A's rule using SCANS (or identify Set B's rule — either works)
2. Confirm it is a negation rule by checking that Set B is the logical opposite
3. For each test shape, ask: "Does this satisfy Set A's rule?"
   • Yes → Set A
   • No → Set B
4. Do not waste time checking "Neither"

---

Worked Example: Straight vs Curved

Set A:

• Box 1: Triangle, square, pentagon (all straight-sided)
• Box 2: Hexagon, rectangle, diamond (all straight-sided)
• Box 3: Star, arrow, triangle (all straight-sided)
• Box 4: Square, square, hexagon (all straight-sided)
• Box 5: Pentagon, triangle, rectangle (all straight-sided)
• Box 6: Diamond, arrow, hexagon, triangle (all straight-sided)

Set B:

• Box 1: Circle, triangle, square (one curved shape)
• Box 2: Semi-circle, hexagon, pentagon (one curved shape)
• Box 3: Circle, circle, rectangle (two curved shapes)
• Box 4: Circle, triangle (one curved)
• Box 5: Semi-circle, circle, pentagon, arrow (two curved)
• Box 6: Circle, diamond, hexagon (one curved)

Rule A: All shapes are straight-sided (no curved shapes). Rule B: At least one shape is curved.