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

Ali Seron Jason

1 month ago

https://www.gyanodhan.com/video/7B4.%20GATE%20CSEIT/Discrete%20Mathematics%201/458.%20Short%20trick%20method%20for%20solving%20propositions%20argument%20based%20problem%20of%20discrete%20mathematics-GATE%202004.mp4

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

Contents

1 Topic: Propositional Logic (Argument Validity) – Shortcut Method
- 1.1 What is the Goal?
2 TRICK: Use the “Assume True Premises, Test False Conclusion” Method
3 Bonus Shortcut Symbols to Remember
4 Truth Value Hints (for implication p → q)
5 Fast Tips:
6 Practice It:

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.

Would you like a PDF notes version, practice problems, or a video explanation with examples for this topic?

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