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

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

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:

1. Assume all premises are TRUE

2. Assume the conclusion is FALSE

3. 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:

1. Assume premises are true:

   • p → q is true

   • q → r is true

2. 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

Truth Value Hints (for implication p → q)

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:

1. ¬p ∨ q

2. ¬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.

Notes on Discrete Mathematics

Discrete Mathematics and Its Applications, Eighth Edition