Specialist Maths - Introduction to Proof Notes
Introduction to Proof
Mathematicians use proofs to ensure rules are always true.
Proofs provide certainty and understanding, unlike scientific theories.
Number Systems
Real Numbers (ℝ): All numbers on the number line.
Natural Numbers (ℕ): Counting numbers, e.g., N={1, 2, 3, 4, 5, …}
Integers (ℤ): Positive and negative natural numbers and 0, e.g., Z={-3, -2, -1, 0, 1, 2, 3…}. Can be positive (ℤ^+) or negative (ℤ^-).
Rational Numbers (ℚ): Any integer divided by another integer, with no common factors other than 1. Can be terminating or recurring decimals.
Irrational Numbers (ℚ' or ℝ/ℚ): Numbers that don't terminate or recur, e.g., surds, \,\pi, e.
Converting Between Decimals and Fractions
Rational numbers can be expressed as terminating or recurring decimals, and vice versa.
To express recurring decimals as a fraction:
Let x = recurring number.
Multiply both sides by a power of 10 based on the recurring part.
Subtract equations and solve for x.
Propositions and Negation
Proposition: A statement that is either true or false.
Negation: The opposite of a statement. If A is true, then ¬A (not A) is false, and vice versa.
Negation of 'A or B': ¬(A \text{ or } B) is the same as ¬A \text{ and } ¬B.
Negation of 'A and B': The negation of the statement ‘A and B’ becomes ‘not A’ or ‘not B’.
Implication and Converse
Implication: If P is true, then Q is true (P ⇒ Q).
P is the hypothesis, Q is the conclusion.
Converse: Switching the hypothesis and conclusion (Q ⇒ P).
Contrapositive
Switch the hypothesis and the conclusion and negating both.
The contrapositive of P ⇒ Q is (not Q) ⇒ (not P).
Equivalence
If P ⇒ Q and Q ⇒ P, then P ⇔ Q (P and Q are equivalent).
P ⇔ Q is true if both P and Q are true or both are false.
Quantifiers
Universal Quantifier (∀): "for all". Example: \forall x ∈ R, x > 7
Existential Quantifier (∃): "there exists". Example: \exists x ∈ R, x > 7
Negation of \forall x ∈ X, P(x) is \exists x ∈ X, ¬P(x).
Negation of \exists x ∈ X, P(x) is \forall x ∈ X, ¬P(x).
Proof by Contradiction
Assume the statement to be proven is false.
Show that this assumption leads to a contradiction.
Conclude that the original statement must be true.
Disproof by Counterexample
A single counterexample is sufficient to disprove a conjecture.
Circle Properties
Theorem 1: The angle at the center of the circle is twice the angle at the circumference subtended on the same arc.
Theorem 2: Angles at the circumference subtended by the same arc are congruent.
Theorem 3: The angle in a semicircle is a right angle (90°).
Cyclic Quadrilateral
A quadrilateral whose vertices all lie on a circle.
Theorem 14: The opposite angles of a cyclic quadrilateral are supplementary (add up to 180°).
Theorem 15: The exterior angle of a cyclic quadrilateral is congruent to the interior opposite angle.
Theorems for Chords
Theorem 4: If a line from the centre bisects a chord, it is perpendicular to the chord, and conversely.
Theorem 6: If two chords intersect inside a circle, then the point of intersection divides each chord into two segments so that the product of the lengths of the segments for both chords is the same.
Tangents and Secants
Theorem 8: A tangent and a radius intersect at right angles.
Theorem 11: The angle formed by a tangent and a chord is congruent to the angle in the alternate segment (Alternate Segment Theorem).
Theorem 12: Tangent-Secant Theorem: If a secant (meeting the circle at A and B) and a tangent (meeting the circle at T) are drawn from an external point P, then PT^2 = PA \cdot PB.
Geometric Proofs using Vectors
A vector has magnitude and direction; a scalar has only magnitude.
If \vec{AB} \parallel \vec{CD}, then \vec{AB} = n\vec{CD} for some scalar n.
If \vec{AB} \perp \vec{CD}, then \vec{AB} \cdot \vec{CD} = 0.
Magnitude: |\vec{a}| = \sqrt{x^2 + y^2}
Dot product: \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos(\theta)
Midpoint M of AB: \vec{OM} = \frac{1}{2}(\vec{a} + \vec{b})