IMAT Mathematics · Free notes

Sets, Intervals and Logical Statements

Sets and Logic is the language the rest of IMAT mathematics is written in. Recent papers give their answers as intervals, write ℝ for all real numbers and ∅ for no solution, and join conditions with ∧ and ∀, so reading these symbols quickly saves time on every algebra question. Direct questions are rare and short: a definition such as the Cartesian product, or a count from a Venn diagram that needs the inclusion and exclusion rule. The logical reasoning puzzles live in their own chapter; this one covers only the mathematical side.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

IMAT home All formulas on one page

Subtopic notes

Formula & revision sheet

1 formulas · 6 reference tables · 7 gotchas across all subtopics — the exam-eve cheat-sheet

Sets, Intervals and Venn Diagrams

Formulas (1)

Counting with Venn diagrams: inclusion and exclusion · Inclusion and exclusion $\begin{aligned} n(A \cup B) &= n(A) + n(B) - n(A \cap B) \\ n(A \cup B \cup C) &= n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \end{aligned}$

Reference tables (3)

Sets, elements, subsets and set notation7 rows

Symbol	Read as	Example
$x \in A$	x is an element of A	$3 \in \{1, 3, 5\}$
$x \notin A$	x is not an element of A	$2 \notin \{1, 3, 5\}$
$A \subseteq B$	A is a subset of B	$\{1, 3\} \subseteq \{1, 3, 5\}$
$A \subset B$	A is a proper subset of B (a subset, not equal to B)	$\mathbb{N} \subset \mathbb{Z}$
$\varnothing$ or $\{\}$	the empty set, with no elements	the solutions of $x^2 = -1$ in $\mathbb{R}$
$n(A)$ or $\|A\|$	the number of elements of A	$n(\{a, b, c\}) = 3$
$\{x \in \mathbb{Z} : x > 0\}$	the integers x such that x is positive	the set $\{1, 2, 3, \ldots\}$

Union, intersection, difference, complement and Cartesian product5 rows

Operation	Symbol	Contains	Example
Union	$A \cup B$	elements in A or in B (or both)	$\{1, 2, 3, 4, 5\}$
Intersection	$A \cap B$	elements in both A and B	$\{3, 4\}$
Difference	$A \setminus B$	elements in A but not in B	$\{1, 2\}$
Complement	$\overline{A}$ or $A^c$	elements of U that are not in A	$\{5, 6\}$
Cartesian product	$A \times B$	all ordered pairs $(a, b)$ with $a \in A$ and $b \in B$	contains $(1, 5)$ but not $(5, 1)$ ; 12 pairs

Intervals of real numbers6 rows

Interval	Inequality	Endpoints
$[a, b]$	$a \le x \le b$	both included (closed)
$(a, b)$	$a < x < b$	both excluded (open)
$[a, b)$	$a \le x < b$	a included, b excluded
$(a, b]$	$a < x \le b$	a excluded, b included
$[a, +\infty)$	$x \ge a$	a included; infinity never is
$(-\infty, a)$	$x < a$	a excluded

Watch out for (4)

An element is not a subset
→ Sets, elements, subsets and set notation
The Cartesian product is a set of pairs, not of products
→ Union, intersection, difference, complement and Cartesian product
Round brackets do not mean a smaller set of whole numbers
→ Intervals of real numbers
Adding the set sizes counts the overlap twice
→ Counting with Venn diagrams: inclusion and exclusion

Propositions, Connectives and Quantifiers

Reference tables (3)

Propositions and the connectives not, and, or, implies5 rows

Connective	Symbol	Read as	True exactly when
Negation	$\neg p$	not p	p is false
Conjunction	$p \wedge q$	p and q	both p and q are true
Disjunction	$p \vee q$	p or q	at least one of p, q is true
Implication	$p \Rightarrow q$	if p then q	it is not the case that p is true and q is false
Biconditional	$p \Leftrightarrow q$	p if and only if q	p and q have the same truth value

Implication, converse, contrapositive and negation4 rows

p	q	p ⇒ q	q ⇒ p (converse)	not q ⇒ not p (contrapositive)
T	T	T	T	T
T	F	F	T	F
F	T	T	F	T
F	F	T	T	T

The third and fifth columns match in every row: an implication and its contrapositive are equivalent. The converse differs.

Quantifiers: for all and there exists4 rows

Statement	In symbols	Its negation
Every x has property P	$\forall x,\ P(x)$	at least one x does not: $\exists x : \neg P(x)$
Some x has property P	$\exists x : P(x)$	no x has it: $\forall x,\ \neg P(x)$
Every real square is at least 0 (true)	$\forall x \in \mathbb{R},\ x^2 \ge 0$	$\exists x \in \mathbb{R} : x^2 < 0$ (false)
Some integer is even and prime (true: 2)	$\exists n \in \mathbb{Z} : n \text{ even} \wedge n \text{ prime}$	every integer is odd or not prime (false)

Watch out for (3)

An implication with a false premise is true
→ Propositions and the connectives not, and, or, implies
The converse is not equivalent
→ Implication, converse, contrapositive and negation
The negation of all is not none
→ Quantifiers: for all and there exists

PYQ weightage by concept

7 concepts · 0 PYQs — where the marks actually sit, so you know what to drill first

Sets, Intervals and Venn Diagrams0 PYQs · 0%

Concept	PYQs	Share
Sets, elements, subsets and set notationfoundation	—	—
Union, intersection, difference, complement and Cartesian productfoundation	—	—
Intervals of real numbersfoundation	—	—
Counting with Venn diagrams: inclusion and exclusionfoundation	—	—

Propositions, Connectives and Quantifiers0 PYQs · 0%

Concept	PYQs	Share
Propositions and the connectives not, and, or, impliesfoundation	—	—
Implication, converse, contrapositive and negationfoundation	—	—
Quantifiers: for all and there existsfoundation	—	—