CDS Mathematics · Sets

Sets and Set Operations

A set is decided by its members alone; union, intersection, difference and complement build new sets from old.

Why this matters

Eight PYQs, one of them HARD. Most test a definition precisely: {0} is not the empty set, {a} is not the element a, the complement of the universal set is empty, and the set of birds on Earth is finite. Work out each set's members before comparing them.

Concept 1 of 1: Members, operations and special sets

Two sets are equal exactly when they have the same members. So list the members first — then union, intersection and difference are just counting which members survive.

Definition

$A - B$ : members of $A$ not in $B$ . $A \cap B$ : in both. $A \cup B$ : in either.
The null set $\phi$ has no members; $\{0\}$ and $\{\phi\}$ each have ONE member, so neither is empty.
An element that is itself a set is different from its member: $\{a\} \ne a$ .
Finite: a definite (possibly huge) count. The real numbers between two values are infinite.
The complement of the universal set is $\phi$ .
Multiples of $7$ and of $5$ intersect in the multiples of $35$ .

Difference

A - B = \{x : x \in A,\ x \notin B\}

Worked example

A = \{2, 4, 6, 8\}

and

B = \{4, 8, 12\}

. Find

A - B

and

B - A

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — Elementary Mathematics · Q10Easy

Example 1 · Sets · Sets and Set Operations

Let

A = \{7, 8, 9, 10, 11, 12\}

and

B = \{7, 10, 14, 15\}

. What is the number of elements in

(A - B)

and

(B - A)

respectively ?

{0} is not empty

The solution set of

x + 5 = 5

\{0\}

, a set with one member. The empty set has none, and

\{\phi\}

has one member, the empty set.

Large is not infinite

However many birds there are, their number is a whole number, so the set is finite. 'Infinite' means no count exists at all.

Drill 7 more on members, operations and special sets

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Members, operations and special sets
Difference
$A - B = \{x : x \in A,\ x \notin B\}$

Watch out for (2)

{0} is not empty
→ Members, operations and special sets
Large is not infinite
→ Members, operations and special sets

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