Unless plus concept – previous year paper discrete mathematics- GATE 2025- If X then Y Unless Z

Unless plus concept – previous year paper discrete mathematics- GATE 2025- If X then Y Unless Z

 “Unless” Concept in Discrete Mathematics – GATE 2025

 Understanding “Unless” in Logic

In propositional logic, “unless” can be rewritten using logical operators.

The statement:

$$\text{“X unless Z”}$$

means:

$$\neg Z\to X$$

(If Z is false, then X must be true.)

Similarly, the statement:

$$\text{“If X then Y unless Z”}$$

can be rewritten as:

$$(Z\vee X)\to Y$$

This means “If Z is true OR X is true, then Y must be true.”

 Equivalent Logical Forms

1. “X unless Z”

   $$X\vee Z$$

   Equivalent to:

   $$\neg Z\to X$$

   (If Z is false, then X must be true.)

2. “If X then Y unless Z”

   $$(Z\vee X)\to Y$$

   Equivalent to:

   $$\neg (Z\vee X)\vee Y$$

   Which simplifies to:

   $$(\neg Z\wedge \neg X)\vee Y$$

 GATE 2025 Previous Year Question on “Unless”

Question:

Which of the following is logically equivalent to:

$$\text{“If X then Y unless Z”}$$

(A) $(X\to Y)\vee Z$
(B) $X\vee (Y\vee Z)$
(C) $(Z\vee X)\to Y$
(D) $\neg Z\vee (X\to Y)$

Solution Approach:

We break down:

$$\text{“If X then Y unless Z”}$$

• “Unless Z” → $X\vee Z$
• “If X then Y” → $X\to Y$

Rewriting:

$$(X\to Y)\vee Z$$

Correct Answer: Option (A)

 Key Takeaways for GATE 2025

“Unless” means OR → $X\vee Z$
Logical equivalence:

• “X unless Z” → $\neg Z\to X$
• “If X then Y unless Z” → $(Z\vee X)\to Y$
  GATE questions often test “unless” using truth tables and logical transformations.

 Need more examples or explanations?

Unless plus concept – previous year paper discrete mathematics- GATE 2025- If X then Y Unless Z

Discrete Mathematics and Its Applications, Eighth Edition

Engineering Mathematics Notes

In Discrete Mathematics and Logical Reasoning, phrases like “If X then Y unless Z” are commonly used in GATE and other competitive exams. Understanding how to logically interpret such statements is key to solving related problems.

“If X then Y unless Z” — What Does It Mean?

This phrase is logically equivalent to:

If X and not Z, then Y
Mathematically:

(X \land \neg Z) \rightarrow Y
]

Breakdown of Components:

• X = a condition

• Y = an outcome that should happen if X holds

• Z = an exception (negates the guarantee of Y when X is true)

“Unless Z” introduces a negation — it means “if Z is not true”.

Example Question (GATE-style):

Statement:
“If it rains, the ground gets wet unless there is a tent.”

Interpretation:
Let:

• $R$: It rains

• $W$: Ground gets wet

• $T$: There is a tent

The statement becomes:

$$(R\wedge \neg T)\to W$$

Previous Year Question Style (GATE):

Q:
“Which of the following correctly expresses: ‘If A occurs, then B occurs unless C’?”

Options:

• (a) $A\to B\vee C$

• (b) $(A\wedge \neg C)\to B$

• (c) $(B\wedge \neg C)\to A$

• (d) $(A\vee C)\to B$

Correct Answer: (b)

Let me know if you’d like practice problems, truth table explanation, or visual diagrams of this logic.