Overview
Discrete Mathematics covers algebraic structures (binary operations, groups) and mathematical logic (statements, connectives, truth tables).
1. Binary Operations
A binary operation on set S satisfies closure: ab ∈ S for all a, b ∈ S. It may also be commutative (ab = ba) or associative ((ab)c = a(bc)).
2. Group Axioms
A set G with operation * is a GROUP if:
1. Closure : a*b ∈ G
2. Associativity : (a*b)*c = a*(b*c)
3. Identity (e) : a*e = e*a = a
4. Inverse : a*a⁻¹ = e
If also commutative → Abelian group.
3. Mathematical Logic
Connectives: ¬ (not), ∧ (and), ∨ (or), → (implies), ↔ (iff)
Tautology : always true
Contradiction: always false
Contrapositive of p→q is ¬q→¬p (logically equivalent)
Worked Examples
Example 1 — Closure
Is subtraction a binary operation on the set of natural numbers ℕ?
2 − 5 = −3 ∉ ℕ → not closed → subtraction is NOT a binary operation on ℕ.
Example 2 — Identity element
*For ab = a + b − 1 on integers, find the identity element.**
a*e = a → a + e − 1 = a → e = 1
Identity element is 1.
Example 3 — Truth table for p → q
p q p→q
T T T
T F F
F T T
F F T
Example 4 — Tautology check
Show p ∨ ¬p is a tautology.
p ¬p p∨¬p
T F T
F T T
Always T → tautology.
Common Mistakes
- Forgetting to verify closure for a binary operation.
- Confusing the identity element with 0 or 1 without checking.
- Writing p→q as F when p is false (it is T).
- Mixing converse (q→p) with contrapositive (¬q→¬p).
Exam Tips & Revision Checklist
- Check all four axioms in order for group problems.
- p→q is false only when p is true and q is false.
- Practised: closure, identity element, truth tables, tautology.
Frequently Asked Questions
A binary operation * on a set S assigns to each pair (a, b) in S a unique element a*b in S (closure).