IMAT Mathematics · Formula sheet

Sets and Logic formulas

1 formula, 6 reference tables and 7 common traps for IMAT Mathematics Sets and Logic, grouped by subtopic.

Sets, Intervals and Venn Diagrams

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}

Sets, elements, subsets and set notation

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 product

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 numbers

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

Common traps

An element is not a subset

2 \in \{1, 2, 3\}

but

\{2\} \subseteq \{1, 2, 3\}

. The symbol

\in

links an object to a set;

\subseteq

links a set to a set. Mixing them makes a statement false.

The Cartesian product is a set of pairs, not of products

A \times B

does not multiply numbers: it lists ordered pairs

(a, b)

. It is not the union or the intersection either. Its size is

n(A) \cdot n(B)

, and the order inside each pair matters.

Round brackets do not mean a smaller set of whole numbers

(1, 4)

is every real number strictly between 1 and 4, including 1.5 and 3.99. It is not the set

\{2, 3\}

. Only set braces

\{\}

list individual elements.

Adding the set sizes counts the overlap twice

n(A) + n(B)

is the union only when the sets are disjoint. Otherwise subtract

n(A \cap B)

once. With three sets, also remember to add back the centre after subtracting the pairs.

Propositions, Connectives and Quantifiers

Learn this subtopic in the notes

Propositions and the connectives not, and, or, implies

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 negation

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 exists

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)

Common traps

An implication with a false premise is true

p \Rightarrow q

only fails when

p

is true and

q

is false. "If 2 > 3 then the moon is cheese" is a true implication. Options that call it false because the parts are false are wrong.

The converse is not equivalent

From "if p then q" you may conclude "if not q then not p", but not "if q then p". Options that reverse the implication without negating both parts are the usual wrong answers.

The negation of all is not none

"Not every student passed" means at least one failed. "No student passed" is a much stronger claim and is not the negation. IMAT options use this confusion.

Sets and Logic formulas

Sets, Intervals and Venn Diagrams

Counting with Venn diagrams: inclusion and exclusion

Sets, elements, subsets and set notation

Union, intersection, difference, complement and Cartesian product

Intervals of real numbers

Common traps

Propositions, Connectives and Quantifiers

Propositions and the connectives not, and, or, implies

Implication, converse, contrapositive and negation

Quantifiers: for all and there exists

Common traps

More IMAT Mathematics formula sheets