Algebra

Introduction

Algebra in Grade 11 is the foundation on which the whole of calculus, coordinate geometry and vector work later stand. It begins with the language of reasoning (logic and truth tables), moves through the structure of the real number system, and then studies the objects that carry the rest of mathematics: relations and functions, sequences and series, matrices and determinants, and finally the complex numbers that complete our number system. Learn the definitions precisely and the standard results by heart, because every later chapter silently assumes them.

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Logic

Logical connectives

A statement (or proposition) is a declarative sentence that is definitely either true or false, but not both. "$7$ is a prime number" is a statement (true); "$x+2=9$" is an open sentence, not a statement, because its truth depends on the value of $x$. Statements are named by letters $p,q,r,\dots$ and each carries a truth value $\text{T}$ (true) or $\text{F}$ (false).

New (compound) statements are built from simple ones using logical connectives:

• Negation $\sim p$ ("not $p$"): true exactly when $p$ is false.
• Conjunction $p \wedge q$ ("$p$ and $q$"): true only when both $p$ and $q$ are true.
• Disjunction $p \vee q$ ("$p$ or $q$"): true when at least one of $p,q$ is true (the inclusive "or").
• Conditional $p \Rightarrow q$ ("if $p$ then $q$"): false only in the single case where $p$ is true and $q$ is false.
• Biconditional $p \Leftrightarrow q$ ("$p$ if and only if $q$"): true exactly when $p$ and $q$ have the same truth value.

For a conditional $p \Rightarrow q$, three related statements are named: the converse $q \Rightarrow p$, the inverse $\sim p \Rightarrow \sim q$, and the contrapositive $\sim q \Rightarrow \sim p$. A conditional is always logically equivalent to its contrapositive, and the converse is equivalent to the inverse.

Truth tables

A truth table lists every possible combination of truth values of the simple statements and computes the truth value of the compound statement. With $n$ simple statements there are $2^n$ rows. The table for the five basic connectives is the one you must be able to reproduce from memory:

$$ \begin{array}{c|c|c|c|c|c|c} p & q & \sim p & p\wedge q & p\vee q & p\Rightarrow q & p\Leftrightarrow q\ \hline \text{T} & \text{T} & \text{F} & \text{T} & \text{T} & \text{T} & \text{T}\ \text{T} & \text{F} & \text{F} & \text{F} & \text{T} & \text{F} & \text{F}\ \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{T} & \text{F}\ \text{F} & \text{F} & \text{T} & \text{F} & \text{F} & \text{T} & \text{T} \end{array} $$

A compound statement that is true in every row is a tautology; one that is false in every row is a contradiction. Two statements are logically equivalent (written $\equiv$) when their final columns are identical.

Worked example. Show that $p \Rightarrow q \equiv \sim p \vee q$.

$$ \begin{array}{c|c|c|c|c} p & q & p\Rightarrow q & \sim p & \sim p \vee q\ \hline \text{T} & \text{T} & \text{T} & \text{F} & \text{T}\ \text{T} & \text{F} & \text{F} & \text{F} & \text{F}\ \text{F} & \text{T} & \text{T} & \text{T} & \text{T}\ \text{F} & \text{F} & \text{T} & \text{T} & \text{T} \end{array} $$

The columns for $p \Rightarrow q$ and $\sim p \vee q$ agree in all four rows, so the statements are equivalent. This is the standard way a conditional is rewritten without an arrow.

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Set identities

A set is a well-defined collection of distinct objects. Recall the union $A \cup B$, the intersection $A \cap B$, the complement $A'$ (relative to a universal set $U$), and the difference $A - B = A \cap B'$. These operations obey algebraic laws that mirror the logic above. The identities most often needed are:

• Commutative: $A \cup B = B \cup A$, $\quad A \cap B = B \cap A$.
• Associative: $A \cup (B \cup C) = (A \cup B) \cup C$, $\quad A \cap (B \cap C) = (A \cap B) \cap C$.
• Distributive: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$, $\quad A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.
• Identity and complement: $A \cup \varnothing = A$, $\ A \cap U = A$, $\ A \cup A' = U$, $\ A \cap A' = \varnothing$.
• De Morgan's laws: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

To prove a set identity you show that each side is a subset of the other, or argue element by element. To prove De Morgan's law $(A \cup B)' = A' \cap B'$, take any $x$:

$$ \begin{aligned} x \in (A \cup B)' &\iff x \notin A \cup B \ &\iff x \notin A \ \text{and}\ x \notin B \ &\iff x \in A' \ \text{and}\ x \in B' \ &\iff x \in A' \cap B'. \end{aligned} $$

Every step is reversible, so the two sets have exactly the same elements and are equal. A Venn diagram can illustrate an identity by shading, but a shading is not a proof; an exam proof argues by elements or by two-way subset inclusion.

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The real number system

Field axioms and order of real numbers

The set $\mathbb{R}$ of real numbers, with addition and multiplication, forms a field. For all $a,b,c \in \mathbb{R}$:

• Closure: $a+b \in \mathbb{R}$ and $ab \in \mathbb{R}$.
• Commutative: $a+b = b+a$ and $ab = ba$.
• Associative: $(a+b)+c = a+(b+c)$ and $(ab)c = a(bc)$.
• Identity: there exist $0$ and $1$ with $a+0 = a$ and $a\cdot 1 = a$.
• Inverse: each $a$ has an additive inverse $-a$ with $a+(-a)=0$; each $a \neq 0$ has a multiplicative inverse $a^{-1}$ with $a,a^{-1}=1$.
• Distributive: $a(b+c) = ab + ac$.

$\mathbb{R}$ is also an ordered field: there is a relation $<$ for which exactly one of $a<b$, $a=b$, $a>b$ holds (trichotomy), and

$$a<b \Rightarrow a+c < b+c, \qquad (a<b \text{ and } c>0) \Rightarrow ac<bc.$$

Multiplying an inequality by a negative number reverses it: if $a<b$ and $c<0$ then $ac>bc$. This one rule is the source of most sign errors, so use it carefully.