Formulas · Mathematics

Sets

3 formulas, each with worked examples. Revision sheet · Practise these formulas

Class 11

Number of elements in a union

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

What each letter means
$n (A \cup B)$	the number of elements in A or B (or both)
$n (A)$	the number of elements in A
$n (B)$	the number of elements in B
$n (A \cap B)$	the number of elements in both A and B

To count the elements of A ∪ B, add the counts of A and of B, then take away the elements in both, which were counted twice.

Why it works

Counting A and then B counts every element in the overlap A ∩ B once in each, so twice. Taking n(A ∩ B) away once leaves every element of the union counted exactly once.

When to use it

Survey problems (how many like tea or coffee, how many play at least one game, how many read neither paper), and Venn diagrams with two circles.

How to use it

Find n(A), n(B) and n(A ∩ B).
Add the first two and take away the overlap.
For "neither", take n(A ∪ B) away from the total.

To remember: Add both circles, take away the lens.

Other forms

Special case $n (A \cup B) = n (A) + n (B)$ When A and B have no element in common (they are disjoint).
To remember itFor three sets: add the three, take away the three overlaps of two, and add back the middle where all three meet.

Worked examples

Story

In a class, 25 students play cricket, 20 play football and 10 play both. How many play at least one of the two?

$25 + 20 - 10 = 35$

Answer: 35

Picture

Two circles of a Venn diagram hold 12 and 15 dots, with 5 dots in the lens where they overlap. How many dots are there in all?

$12 + 15 - 5 = 22$

Answer: 22

Direct

n(A) = 20, n(B) = 28 and n(A ∩ B) = 8. Find n(A ∪ B).

$20 + 28 - 8 = 40$

Answer: 40

Reverse

In a group of 50 people, 35 speak Hindi and 25 speak English, and everyone speaks at least one of them. How many speak both?

Try it first, then show the working

$35 + 25 - 50 = 10$

Answer: 10

Exam

Of 400 people, 250 read newspaper H, 200 read newspaper T and 100 read both. How many read neither?

Try it first, then show the working

$250 + 200 - 100 = 350$
So 400 − 350 = 50 read neither paper.

Answer: 50

Common mistake: Adding n(A) and n(B) without removing the overlap, which counts the people in both groups twice.

Class 11

Number of elements in a Cartesian product

n (A \times B) = n (A) \times n (B)

What each letter means
$n (A \times B)$	the number of ordered pairs in A × B
$n (A)$	the number of elements in A
$n (B)$	the number of elements in B

A × B is the set of all ordered pairs (a, b) with a from A and b from B, and it has n(A) × n(B) of them.

Why it works

Each of the n(A) first entries can be paired with each of the n(B) second entries: n(A) rows of n(B) pairs, which is the counting principle again.

When to use it

Counting ordered pairs, points on a grid, and the pairs available to a relation from A to B.

How to use it

Count the elements of A and of B.
Multiply.

To remember: Every first with every second: multiply.

Other forms

Special case $n (A \times A) = n (A)^{2}$ When B = A.
To remember itFor three sets, n(A × B × C) = n(A) × n(B) × n(C).

Worked examples

Story

A café offers 4 drinks and 5 snacks. How many (drink, snack) pairs can be ordered?

$4 \times 5 = 20$

Answer: 20

Picture

The points (x, y) with x from {1, 2, 3} and y from {1, 2} make a grid. How many points are there?

$3 \times 2 = 6$

Answer: 6

Direct

A has 3 elements and B has 4. How many ordered pairs are in A × B?

$3 \times 4 = 12$

Answer: 12

Reverse

A × B has 15 elements and A has 5. How many elements does B have?

Try it first, then show the working

$5 \times 3 = 15$

Answer: 3

Exam

A × B = {(p, q), (p, r), (m, q), (m, r)}. Find A and B.

Try it first, then show the working

The first entries give A = {p, m}; the second entries give B = {q, r}.
Check the count:
$2 \times 2 = 4$

Answer: A = {p, m}, B = {q, r}

Common mistake: Adding the counts, or treating (a, b) and (b, a) as the same pair; order matters in an ordered pair.

Class 11

Number of relations from A to B

N = 2^{pq}

What each letter means
$N$	the number of different relations from A to B
$p$	the number of elements in A
$q$	the number of elements in B

A relation from A to B is any subset of A × B. Since A × B has pq pairs, and each pair is either in the relation or not, there are 2^(pq) relations.

Why it works

Build a relation by going through the pq pairs and deciding for each one: in or out. That is 2 choices, pq times, so 2 × 2 × … × 2 = 2^(pq) ways.

When to use it

Counting the possible relations between two finite sets, and the subsets of any finite set.

How to use it

Find the number of pairs, pq.
Raise 2 to that power.

To remember: Every pair: in or out.

Other forms

Special case $N = 2^{p^{2}}$ When the relation is on one set A (B = A, so q = p).
To remember itThe empty set and the whole of A × B are both relations: the smallest and the largest.

Worked examples

Story

Three students may each join a chess club or not. How many different membership lists are possible?

Each of the 3 × 1 = 3 (student, club) pairs is in or out.
$2^{3 \times 1} = 8$

Answer: 8

Picture

Each square of a 2 by 2 grid can be shaded or left blank. How many patterns are there?

Each of the 2 × 2 = 4 squares is shaded or not.
$2^{2 \times 2} = 16$

Answer: 16

Direct

A has 2 elements and B has 3. How many relations are there from A to B?

$2^{2 \times 3} = 64$

Answer: 64

Reverse

There are 512 relations from A to B, and A has 3 elements. How many elements does B have?

Try it first, then show the working

$2^{3 \times 3} = 512$

Answer: 3

Exam

A = {1, 2} and B = {3, 4}. How many relations are there from A to B? Give one with exactly two pairs.

Try it first, then show the working

A × B has 4 pairs, so there are 2⁴ = 16 relations.
One with two pairs is {(1, 3), (2, 4)}; there are ⁴C₂ = 6 such relations.
$2^{4} = 16$

Answer: 16; for example {(1, 3), (2, 4)}

Common mistake: Writing 2 × pq or pq², instead of 2 to the power pq.