Short trick method for solving propositions argument based problem of discrete mathematics-GATE 2025.

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Here’s a short-trick method for solving Propositions/Argument-Based Problems in Discrete Mathematics – highly useful for GATE 2025 CSE/IT aspirants.

🧠 Topic: Propositional Logic (Argument Validity) – Shortcut Method

🎯 What is the Goal?

You are often asked:

“Is this argument valid?”
Or
“Which conclusion logically follows from the given premises?”

Instead of using truth tables (which are time-consuming), use this shortcut method:

🔍 TRICK: Use the “Assume True Premises, Test False Conclusion” Method

✅ Step-by-Step:

Assume all premises are TRUE
Assume the conclusion is FALSE
Try to find a truth assignment (T/F for each variable) that satisfies both.
- If you can, argument is invalid.
- If you cannot, argument is valid.

📘 Example Question (GATE Style):

Given:

Premise 1: p → q
Premise 2: q → r
Conclusion: p → r

Is the argument valid?

🔍 Apply the Trick:

Assume premises are true:
- p → q is true
- q → r is true
Assume conclusion is false:
- p → r is false ⟹ p = T, r = F

Now check:

p = T
r = F
For p → q to be true and p = T, then q = T
For q → r to be true and q = T, then r = T ❌ (but we assumed r = F)

❌ Contradiction. So no such assignment exists.

✅ Conclusion: Argument is Valid

⚡ Bonus Shortcut Symbols to Remember

Symbol	Meaning
∧	AND
∨	OR
→	IMPLIES
↔	BICONDITIONAL (IFF)
¬	NOT (Negation)

✨ Truth Value Hints (for implication `p → q`)

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

Only false when p = T and q = F

📌 Fast Tips:

Conjunction (p ∧ q) is false if any one is false
Disjunction (p ∨ q) is true if any one is true
Implication (p → q) is only false when p = T and q = F

🧪 Practice It:

👉 Try this:

Premises:

¬p ∨ q
¬q ∨ r

Conclusion:
¬p ∨ r

Try using the same trick and you’ll quickly conclude: the argument is valid.

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