Mathematical Logic

Let's start with something that's basically the grammar of mathematics — mathematical logic. You know how every sentence in English can be true, false, or somewhere in between? In maths, we only care about sentences that are definitively true or false. No grey areas. No opinions. Just cold, hard truth values. And then we combine those sentences using words like 'and', 'or', 'not', 'if...then', and 'if and only if' to build more complex statements. Now, you might be thinking — why does this matter? Well, mathematical logic is the foundation of everything in computer science. Every time you type a condition in code — if (x > 5 && y < 10) — you're using logic. Every search engine, every AI model, every digital circuit — it all runs on logic gates that are built from these same principles. In fact, the computer you're reading this on uses millions of tiny logic gates that implement exactly the operations we're about to study. So when you master this chapter, you're not just learning maths — you're learning how computers think. The syllabus starts with understanding what a statement actually is, because not every sentence qualifies. Then we move to logical connectives — the glue that joins simple statements into compound ones. You'll meet the five main operators: conjunction (AND, written as ∧), disjunction (OR, ∨), negation (NOT, ∼), implication (IF...THEN, →), and biconditional (IF AND ONLY IF, ↔). Each has its own truth table — a systematic way to list every possible combination of truth values and see what the result is. Truth tables are your best friend in this chapter. They're predictable, logical, and once you understand the pattern, you can handle any compound statement. Then comes the fun part — tautologies (statements that are always true, like 'it is raining or it is not raining'), contradictions (always false), and contingencies (sometimes true, sometimes false). You'll also learn about logical equivalence — two statements that have the same truth table. This is incredibly powerful because it lets you replace a complicated statement with a simpler one that means the same thing. De Morgan's laws are the star players here: ∼(p ∧ q) ≡ ∼p ∨ ∼q and ∼(p ∨ q) ≡ ∼p ∧ ∼q. Think of them as the 'distribution of negation' rules. Then there's the algebra of statements — commutative, associative, distributive, identity, complement laws — it's like the algebra you already know, but with logic operators instead of plus and times. You'll also encounter quantifiers: 'for all' (∀) and 'there exists' (∃). These let you make statements about entire sets of objects. 'All humans are mortal' uses the universal quantifier. 'There exists a prime number greater than 100' uses the existential quantifier. Negating quantifiers can be tricky — the negation of 'all are' is 'some are not', and vice versa. Converse, inverse, and contrapositive are about flipping implications around. If the original statement is 'if p then q', the converse is 'if q then p', the inverse is 'if not p then not q', and the contrapositive is 'if not q then not p'. Here's a cool trick: a statement and its contrapositive are logically equivalent. So if you're stuck proving something, try proving its contrapositive instead. We also cover duality — where you swap ∧ with ∨ and T with F — and the principle of duality says that if a statement is a tautology, its dual is also a tautology. Finally, you'll see how all of this applies to switching circuits. Every logic gate in a circuit corresponds to a logical operation. AND gates, OR gates, NOT gates — they're the hardware implementation of what we're studying here. By the end of this chapter, you'll be able to take a real-world condition, translate it into symbolic logic, simplify it, and even design a circuit for it. That's not just maths — that's engineering.