Mathematical Logic
Easy Overview
Ever wondered how computers decide between true and false? That's logic at work. This chapter is basically about figuring out whether statements are true or false, and how to combine them in different ways. It's the grammar of maths — once you get the rules, everything clicks.
Statements and Truth Values
A statement is just a sentence that is either true or false — no in-between. 'Mumbai is in India' is a statement (true). 'This is fun' is NOT a statement because it's an opinion. Simple. Every statement gets a truth value: T or F.
Logical Connectives — And, Or, Not
Think of connectives like seasoning. 'And' (∧) means both need to be true. 'Or' (∨) means at least one is true. 'Not' (∼) just flips the truth. If I say 'I'll bring pizza AND garlic bread', you only get happy if both show up. But 'I'll bring pizza OR garlic bread'? You win either way.
Truth Tables
Truth tables are just cheat sheets. You list every possible combo of true/false for your statements, then figure out the result. For two statements, there are 4 combos. For three? 8. It's like a multiplication table — predictable once you know the rule.
Implication and Biconditional
'If it rains, the ground gets wet' — that's an implication (→). The first part is the condition, the second is the result. The only time it's false is when the condition is true but the result is false. The biconditional (↔) is a two-way street: 'I go out IF AND ONLY IF it's sunny'.
Tautologies and Contradictions
A tautology is always true, no matter what. Like 'it's either raining or not raining' — can't argue with that. A contradiction is always false, like 'it's raining AND it's not raining'. Spotting these saves time in exams — they're the shortcuts of logic.
Key Points
- •A statement must be objectively true or false — no opinions allowed
- •∧ (and) — both T → T; ∨ (or) — at least one T → T
- •∼ flips truth: ∼T = F, ∼F = T
- •Implication p → q is false ONLY when p is T and q is F
- •Biconditional p ↔ q is T when both have same truth value
- •Tautology = always T; Contradiction = always F
- •Truth tables = systematic way to check all possibilities
Practice Questions
- Prepare a truth table for (p ∧ q) ∨ (∼p) and identify if it's a tautology.
- Write the truth value of 'If 3 > 5 then 7 > 2'.
- Using logical equivalence, show that ∼(p ∨ q) ≡ ∼p ∧ ∼q.
- Express 'If you work hard then you will succeed' in symbolic form and write its converse.