Mathematical Notation and Logic

Mathematics uses a concise symbolic language to express complex ideas. Understanding this notation is essential for reading and writing formal proofs.

Logical Operators

The building blocks of mathematical logic are propositions and operators:

• Negation ($\neg P$): “Not $P$“.
• Conjunction ($P \land Q$): ”$P$ and $Q$“.
• Disjunction ($P \lor Q$): ”$P$ or $Q$“.
• Implication ($P \implies Q$): “If $P$, then $Q$“.
• Equivalence ($P \iff Q$): ”$P$ if and only if $Q$“.

def logical_implication(p, q):
  # P => Q is equivalent to (not P) or Q
  return (not p) or q

print("Truth Table for P => Q")
print("P | Q | P => Q")
for p in [True, False]:
  for q in [True, False]:
      print(f"{p} | {q} | {logical_implication(p, q)}")

Quantifiers

Quantifiers allow us to make statements about sets of elements:

• Universal Quantifier ($\forall$): “For all”.
• Existential Quantifier ($\exists$): “There exists”.

numbers = [1, 2, 3, 4, 5]

# For all x in numbers, x > 0
all_positive = all(x > 0 for x in numbers)
print(f"Universal (forall x > 0): {all_positive}")

# There exists x in numbers such that x is even
exists_even = any(x % 2 == 0 for x in numbers)
print(f"Existential (exists x even): {exists_even}")

Set Notation

• $x \in A$: ”$x$ is an element of $A$“.
• $A \subseteq B$: ”$A$ is a subset of $B$“.
• $A \cap B$: Intersection.
• $A \cup B$: Union.