6.1 The Language of Proof
SL 1.6
- the words theorem, proof, definition, proposition, lemma and corollary
- the symbols used to write mathematical statements, including quantifiers \(\forall\) and \(\exists\)
- the difference between an equation (\(=\)) and an identity (\(\equiv\))
- how to find unknown constants in an identity
Words used in proof
| Word | Meaning |
|---|---|
| theorem | an important mathematical statement that has been shown to be true |
| proof | a logical argument that shows, without any doubt, that a statement is true |
| definition | an exact statement of what a word or symbol means |
| proposition | a true statement that is less important than a theorem |
| lemma | a smaller result proved mainly to help prove a theorem |
| corollary | a result that follows quickly from a theorem that has just been proved |
| conjecture | a statement that seems true but has not been proved (see 2.1) |
A proof usually ends with a short closing statement, such as “LHS = RHS”, “as required”, the word “shown”, or the letters QED (Latin quod erat demonstrandum, “which was to be shown”).
Symbols
| Symbol | Meaning | Symbol | Meaning |
|---|---|---|---|
| \(\mathbb{N}\) | natural numbers \(\{0, 1, 2, \ldots\}\) | \(\Rightarrow\) | implies |
| \(\mathbb{Z}\) | integers | \(\Leftrightarrow\) | if and only if (iff) |
| \(\mathbb{Z}^+\) | positive integers \(\{1, 2, 3, \ldots\}\) | \(\in\) | is an element of |
| \(\mathbb{Q}\) | rational numbers | \(\notin\) | is not an element of |
| \(\mathbb{R}\) | real numbers | \(\therefore\) | therefore |
| \(\mathbb{C}\) | complex numbers | \(\because\) | because |
| \(\forall\) | for all | \(\exists\) | there exists (at least one) |
| \(\cap\) | intersection (“and”) | \(\cup\) | union (“or”) |
| \(\subset\) | is a subset of | \(\setminus\) | set minus (remove) |
The IB uses \(\mathbb{N} = \{0, 1, 2, \ldots\}\), so \(0 \in \mathbb{N}\).
Quantifiers
Two symbols tell us how many values a statement is about.
- Universal quantifier \(\forall\) (“for all”): \(\;\forall x \in \mathbb{R},\; x^2 \ge 0\) means “for every real number \(x\), \(x^2 \ge 0\)”. It is true only if it works for every \(x\).
- Existential quantifier \(\exists\) (“there exists”): \(\;\exists x \in \mathbb{Z}\) such that \(x^2 = 9\) means “there is at least one integer whose square is 9”. It is true if it works for at least one \(x\).
To disprove a “for all” statement, one counterexample is enough (see 6.2).
Example 1 Write each statement in words, and say whether it is true.
(a) \(\exists\, x \in \mathbb{Z}\) such that \(2x = 7\) \(\qquad\) (b) \(\forall\, n \in \mathbb{Z}^+,\; n^2 \ge n\)
(c) \(\forall\, x \in \mathbb{R},\; \exists\, y \in \mathbb{R}\) such that \(y > x\) \(\qquad\) (d) \(\exists\, q \in \mathbb{Q}\) such that \(q^2 = 2\)
(a) “There is an integer \(x\) with \(2x = 7\).” False: the only solution is \(3.5\), which is not an integer.
(b) “For every positive integer \(n\), \(n^2 \ge n\).” True: \(n^2 - n = n(n - 1) \ge 0\) when \(n \ge 1\).
(c) “For every real number \(x\), there is a real number \(y\) bigger than \(x\).” True: take \(y = x + 1\).
(d) “There is a rational number whose square is 2.” False: \(\sqrt{2}\) is irrational (proved in 6.3).
Example 2 Write each statement using mathematical symbols.
(a) For every real number \(a\), \(a^2\) is not negative.
(b) \(b\) is a natural number greater than or equal to 10 but less than 40.
(c) There is a negative integer \(c\) whose cube is \(-8\).
(d) For every rational number \(p\), there exists an integer greater than \(p\).
(a) \(\forall\, a \in \mathbb{R},\; a^2 \ge 0\)
(b) \(b \in \mathbb{N},\; 10 \le b < 40\)
(c) \(\exists\, c \in \mathbb{Z},\; c < 0\), such that \(c^3 = -8\)
(d) \(\forall\, p \in \mathbb{Q},\; \exists\, m \in \mathbb{Z}\) such that \(m > p\)
Equations and identities
An equation is true only for some values of the variable. An identity is true for all values. We use the sign \(\equiv\) to show an identity.
\[ \begin{gathered} 2x + 1 = 7 \quad (\text{true only when } x = 3) \\ (x + 1)^2 \equiv x^2 + 2x + 1 \quad (\text{true for every } x) \end{gathered} \]
Example 3 Which of these are identities?
(a) \(3(x + 2) - x = 2(x + 3)\) \(\qquad\) (b) \(4x - 1 = 3(x + 1)\)
(c) \((x - 3)^2 = x^2 - 9\) \(\qquad\) (d) \((x - 3)(x + 3) = x^2 - 9\)
(a) LHS \(= 2x + 6 =\) RHS for every \(x\): identity.
(b) Only true when \(x = 4\): an equation, not an identity.
(c) \((x - 3)^2 = x^2 - 6x + 9\), which equals \(x^2 - 9\) only when \(x = 3\): not an identity.
(d) Expanding the left side gives exactly \(x^2 - 9\): identity.
Finding constants in an identity
If two polynomials are identical, the coefficients of each power of \(x\) must be equal. This is called equating coefficients. (You used it for partial fractions in 3.1.)
Example 4 Given that \(x^2 + 8x + 3 \equiv (x + a)^2 + b\), find the values of \(a\) and \(b\).
Expand the right-hand side: \((x + a)^2 + b = x^2 + 2ax + a^2 + b\).
\[ \text{coefficients of } x:\; 2a = 8 \;\Longrightarrow\; a = 4 \qquad\quad \text{constants}:\; a^2 + b = 3 \;\Longrightarrow\; b = 3 - 16 = -13 \]
So \(x^2 + 8x + 3 \equiv (x + 4)^2 - 13\).
Common errors
| ✗ Wrong | ✓ Right | Why |
|---|---|---|
| Treating \(\forall\) and \(\exists\) as the same | “all” vs “at least one” | They give completely different statements. |
| \(0 \notin \mathbb{N}\) | \(0 \in \mathbb{N}\) in the IB | The IB includes 0 in the natural numbers. |
| Checking an identity with one value of \(x\) | show it for all \(x\) | One value can only disprove an identity. |
| \(\equiv\) and \(=\) used the same way | \(\equiv\) means “for all values” | An equation is solved; an identity is proved. |
Exercises
1 Write in words, and state whether each is true:
(a) \(\forall\, x \in \mathbb{R},\; x + 0 = x\)
(b) \(\exists\, n \in \mathbb{N}\) such that \(n < 0\)
(a) “For every real number \(x\), \(x + 0 = x\).” True.
(b) “There is a natural number less than 0.” False: the smallest natural number is \(0\).
2 Write in symbols: “There exists a real number \(x\) such that \(x^2 = x\).”
\(\exists\, x \in \mathbb{R}\) such that \(x^2 = x\). (True: \(x = 0\) or \(x = 1\).)
3 Write in symbols: “The square of every integer is a natural number.”
\(\forall\, n \in \mathbb{Z},\; n^2 \in \mathbb{N}\).
4 Is \(\forall\, x \in \mathbb{R},\; \sqrt{x^2} = x\) true? Explain.
False. For \(x = -3\): \(\sqrt{9} = 3 \neq -3\). (The correct statement is \(\sqrt{x^2} = |x|\).)
5 Decide whether \((2x + 1)^2 - (2x - 1)^2 = 8x\) is an identity.
\((4x^2 + 4x + 1) - (4x^2 - 4x + 1) = 8x\) for every \(x\). Yes, it is an identity.
6 Decide whether \(\dfrac{x^2 - 4}{x - 2} = x + 2\) is an identity for all real \(x\).
It is true for every \(x \neq 2\), but at \(x = 2\) the left side is undefined. So it is an identity only on \(\mathbb{R} \setminus \{2\}\).
7 Given that \(2x^2 - 12x + 5 \equiv 2(x - p)^2 + q\), find \(p\) and \(q\).
\(2(x - p)^2 + q = 2x^2 - 4px + 2p^2 + q\). So \(-4p = -12\), \(p = 3\), and \(2(9) + q = 5\), \(q = -13\).
8 Find \(a\), \(b\) and \(c\) such that \(3x^2 + 5x - 2 \equiv a(x + 1)^2 + b(x + 1) + c\).
RHS \(= ax^2 + (2a + b)x + (a + b + c)\). So \(a = 3\); \(2a + b = 5 \Rightarrow b = -1\); \(a + b + c = -2 \Rightarrow c = -4\).
9 Explain the difference between \(\forall\, x \;\exists\, y\) such that \(x + y = 0\), and \(\exists\, y \;\forall\, x\), \(x + y = 0\) (with \(x, y \in \mathbb{R}\)).
The first says: for each \(x\) we can find a \(y\) (it may depend on \(x\)) — true, \(y = -x\).
The second says: one single \(y\) works for every \(x\) — false, no fixed \(y\) gives \(x + y = 0\) for all \(x\). The order of the quantifiers matters.
10 Write down one number that belongs to \(\mathbb{Q}\) but not to \(\mathbb{Z}\), and one that belongs to \(\mathbb{R}\) but not to \(\mathbb{Q}\).
For example \(\dfrac{1}{2} \in \mathbb{Q} \setminus \mathbb{Z}\), and \(\sqrt{2}\) or \(\pi \in \mathbb{R} \setminus \mathbb{Q}\).