2.1 Real Numbers, Sets, Logic, Binary Operations & Groups

Real Numbers, Sets, Logic, Binary Operations & Groups

The first chapters of Intermediate Mathematics Part I set up the language that the rest of the course uses: number systems, sets, logic and algebraic structures. These topics produce short recall and reasoning questions, so they are worth securing early.

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1. The Real Number System

Every terminating or recurring decimal is rational. To convert a recurring decimal, multiply by a power of 10 and subtract:

$x = 0.\overline{45} \implies 100x = 45.\overline{45} \implies 99x = 45 \implies x = \frac{45}{99} = \frac{5}{11}$

Common trap: $\pi$ is irrational, so $\frac{22}{7}$ is only an approximation. Likewise $\sqrt{4} = 2$ is rational even though it is written with a root sign.

Field and Order Properties of $\mathbb{R}$

For all real $a$, $b$, $c$:

• Closure: $a + b$ and $ab$ are real.
• Commutative: $a + b = b + a$, $ab = ba$.
• Associative: $(a + b) + c = a + (b + c)$, $(ab)c = a(bc)$.
• Identities: $0$ for addition, $1$ for multiplication.
• Inverses: $-a$ for addition; $\frac{1}{a}$ for multiplication when $a \neq 0$.
• Distributive: $a(b + c) = ab + ac$.
• Order (trichotomy): exactly one of $a < b$, $a = b$, $a > b$ holds.
• Transitive: $a < b$ and $b < c$ imply $a < c$.
• Multiplicative order: if $a < b$ and $c > 0$ then $ac < bc$; if $c < 0$ the inequality reverses, $ac > bc$.

The last rule is the source of many errors when solving inequalities: multiplying or dividing by a negative number flips the sign.

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2. Sets and Set Operations

A set is a well-defined collection of distinct objects. It can be written in tabular form, $A = \{2, 4, 6\}$, or set-builder form, $A = \{x \mid x \text{ is an even natural number}, x \le 6\}$.

• Subset: $A \subseteq B$ if every element of $A$ is in $B$. A set with $n$ elements has $2^n$ subsets and $2^n - 1$ proper subsets.
• Power set: $P(A)$ is the set of all subsets of $A$, so $n(P(A)) = 2^n$. For $A = \{a, b, c\}$, $n(P(A)) = 8$.
• Union, intersection, difference, complement: $A \cup B$, $A \cap B$, $A - B = A \cap B'$, and $A' = U - A$.
• Disjoint sets: $A \cap B = \varnothing$.

Laws of Set Algebra

Counting with Venn Diagrams

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$

Worked example: In a class of 60 students, 35 study Physics, 30 study Chemistry and 12 study both. Then $n(P \cup C) = 35 + 30 - 12 = 53$, so $60 - 53 = 7$ study neither. The number studying Physics only is $35 - 12 = 23$.

Ordered Pairs and Cartesian Products

$(a, b) = (c, d)$ only if $a = c$ and $b = d$. The Cartesian product $A \times B = \{(a, b) \mid a \in A, b \in B\}$ has $n(A) \cdot n(B)$ elements. A relation from $A$ to $B$ is any subset of $A \times B$, so there are $2^{n(A)n(B)}$ possible relations. A function is a relation in which every element of $A$ appears as a first entry exactly once (functions are developed in Section 5.1).

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3. Propositional Logic

A proposition is a statement that is either true or false. The basic connectives are:

The conditional $p \to q$ is false only when $p$ is true and $q$ is false. A false premise makes the conditional true ("vacuously true").

Converse, Inverse and Contrapositive

For the conditional $p \to q$:

• Converse: $q \to p$
• Inverse: $\sim p \to \sim q$
• Contrapositive: $\sim q \to \sim p$

A conditional is logically equivalent to its contrapositive; the converse and inverse are equivalent to each other but not to the original. Example: "If a number is divisible by 4, it is even" is true; its converse "If a number is even, it is divisible by 4" is false (6 is a counterexample); its contrapositive "If a number is not even, it is not divisible by 4" is true.

Tautologies, Contradictions and Quantifiers

• A tautology is always true, such as $p \vee \sim p$ or $(p \wedge q) \to p$.
• A contradiction is always false, such as $p \wedge \sim p$.
• A contingency is sometimes true and sometimes false.
• Logical De Morgan laws: $\sim(p \wedge q) \equiv \sim p \vee \sim q$ and $\sim(p \vee q) \equiv \sim p \wedge \sim q$.
• Quantifiers: the negation of "for all $x$, $P(x)$" is "there exists an $x$ such that not $P(x)$".

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4. Binary Operations and Groups

A binary operation $*$ on a set $S$ combines any two elements of $S$ to give an element of $S$ (closure is built into the definition). A non-empty set $G$ with a binary operation $*$ is a group if:

1. Closure: $a * b \in G$ for all $a, b \in G$.
2. Associativity: $(a * b) * c = a * (b * c)$.
3. Identity: there is $e \in G$ with $a * e = e * a = a$.
4. Inverses: every $a$ has $a^{-1} \in G$ with $a * a^{-1} = a^{-1} * a = e$.

If, in addition, $a * b = b * a$ for all elements, the group is abelian (commutative). A set with only closure and associativity is a semi-group; a semi-group with identity is a monoid.

Worked example: On $\mathbb{R}$ define $a * b = a + b + 3$. The identity $e$ satisfies $a + e + 3 = a$, so $e = -3$. The inverse of $a$ satisfies $a + a^{-1} + 3 = -3$, so $a^{-1} = -6 - a$. The operation is associative and commutative, so $(\mathbb{R}, *)$ is an abelian group.

In any group, the identity is unique, each element has exactly one inverse, the cancellation laws hold, and $(ab)^{-1} = b^{-1}a^{-1}$.