Sets and Binary Operations: JAMB Mathematics Notes

A step-by-step guide to set notation and operations, Venn-diagram counting, and custom binary operations including identity, inverse, and commutativity.

Set Concepts & Operations

A set is a well-defined collection of objects. We usually write its elements inside braces. For example,

$A = {2, 4, 6, 8}$ .

The statement $4 \in A$ means “4 belongs to $A$ ,” while $5 \in / A$ means “5 does not belong to $A$ .”

Two sets are equal when they contain exactly the same elements. They are equivalent when they contain the same number of elements, even if the elements themselves are different.

A set $A$ is a subset of $B$ , written $A \subseteq B$ , if every element of $A$ is also in $B$ . A proper subset is a subset that is not equal to the whole set.

If a set has $n$ distinct elements, it has $2^{n}$ subsets in total. Those subsets form the power set.

The main set operations are:

Union: $A \cup B$ contains everything in $A$ or $B$ , without repeating elements.
Intersection: $A \cap B$ contains only elements common to both.
Difference: $A ∖ B$ contains elements in $A$ that are not in $B$ .
Complement: $A^{'}$ contains elements of the universal set $U$ that are not in $A$ .
Symmetric difference: elements in exactly one of the two sets, not in both.

Remember that repeated entries do not change a set. For example, ${1, 1, 2, 2, 3} = {1, 2, 3}$ .

Example 1: Let $A = {1, 3, 5}$ and $B = {2, 4, 6}$ . Are the sets equal or equivalent?

They are not equal because their elements differ. Each has three elements, so they are equivalent.

Example 2: Let $A = {1, 2, 3, 4}$ and $B = {3, 4, 5, 6}$ . Find $A \cup B$ .

Collect every distinct element appearing in either set:

$A \cup B = {1, 2, 3, 4, 5, 6}$ .

Example 3: Using the same sets, find $A \cap B$ .

The common elements are 3 and 4, so

$A \cap B = {3, 4}$ .

Example 4: Using the same sets, find $A ∖ B$ .

Keep the elements that belong to $A$ but not $B$ :

$A ∖ B = {1, 2}$ .

Example 5: Let $U = {1, 2, 3, 4, 5, 6, 7, 8}$ and $A = {2, 4, 6, 8}$ . Find $A^{'}$ .

Remove the elements of $A$ from the universal set:

$A^{'} = {1, 3, 5, 7}$ .

Example 6: How many subsets does $S = {a, b, c, d}$ have?

There are four elements, so the number of subsets is

$2^{4} = 16$ .

Only one of these subsets is the whole set itself, so there are $16 - 1 = 15$ proper subsets if the empty set is counted as a proper subset.

Example 7: Let $P = {1, 2, 3, 4}$ and $Q = {2, 4, 6}$ . Find the elements that belong to exactly one of $P$ and $Q$ .

$P ∖ Q = {1, 3}$ and $Q ∖ P = {6}$ .

Their union is ${1, 3, 6}$ .

Venn Diagrams & Set Counting

A Venn diagram turns set relationships into regions. Before calculating, decide what each region means. In a two-set diagram:

the overlap is $A \cap B$ ;
the two circles together are $A \cup B$ ;
outside both circles is “neither.”

For two sets,

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

We subtract the intersection once because adding $n (A)$ and $n (B)$ counts the overlap twice.

If the universal set has $n (U)$ elements, then

$n ((A \cup B)^{'}) = n (U) - n (A \cup B)$ .

For three sets,

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

The final three-way intersection is added back because the earlier subtraction removes it too many times.

Example 8: In a class of 40 students, 25 study Physics, 20 study Chemistry, and 10 study both. How many study at least one subject?

$n (P \cup C) = 25 + 20 - 10 = 35$ .

So 35 students study at least one.

Example 9: Using Example 8, how many study neither subject?

There are 40 students in total and 35 in the union.

Therefore $40 - 35 = 5$ study neither.

Example 10: In a group, $n (A) = 20$ , $n (B) = 30$ , and $n (A \cup B) = 40$ . Find $n (A \cap B)$ .

Use $40 = 20 + 30 - n (A \cap B)$ .

Hence $n (A \cap B) = 10$ .

Example 11: A class has 65 students. 45 offer Physics, 38 offer Chemistry, and every student offers at least one. How many offer both?

Because every student is in the union, $n (P \cup C) = 65$ .

Thus $65 = 45 + 38 - n (P \cap C)$ , so $n (P \cap C) = 18$ .

Example 12: In a survey of 100 people, 60 like tea, 50 like coffee, and 20 like both. Find the probability that a randomly chosen person likes neither.

The union is $60 + 50 - 20 = 90$ .

So 10 like neither, and the probability is $10/100 = 1/10$ .

Example 13: In a group, 30 study Mathematics, 25 Physics, and 20 Chemistry. Pairwise intersections are 12, 10, and 8 respectively, and 5 study all three. Find the number studying at least one.

Use inclusion-exclusion:

$30 + 25 + 20 - 12 - 10 - 8 + 5 = 50$ .

So 50 people study at least one of the three subjects.

Binary Operations

A binary-operation symbol such as $*$ , $\circ$ , or $⋆$ does not automatically mean ordinary multiplication. The question defines what the symbol means. Your first job is to copy that rule and substitute the two inputs carefully.

For example, if

$a * b = a + 2 b$ ,

then $3 * 4$ means

$3 + 2 (4) = 11$ .

When operations are nested, work from the innermost brackets outward, just as you do with ordinary algebra.

An identity element $e$ leaves every input unchanged. For an operation $*$ , it must satisfy the required identity condition, typically

$a * e = e * a = a$ .

An inverse of $a$ is an element that combines with $a$ to give the identity.

An operation is commutative if changing the order never changes the result:

$a * b = b * a$ .

It is associative if regrouping never changes the result:

$(a * b) * c = a * (b * c)$ .

These are different properties; one does not automatically imply the other.

Example 14: If $x * y = x + y^{2}$ , find $2 * 3$ .

Substitute $x = 2$ and $y = 3$ :

$2 * 3 = 2 + 3^{2} = 11$ .

Example 15: If $p * q = pq + p - q$ , find $2 * 4$ .

$2 * 4 = (2) (4) + 2 - 4 = 8 + 2 - 4 = 6$ .

Example 16: If $x * y = x + 2 y$ , find $2 * (3 * 4)$ .

Do the inner operation first:

$3 * 4 = 3 + 2 (4) = 11$ .

Then $2 * 11 = 2 + 2 (11) = 24$ .

Example 17: If $a \circ b = a^{b}$ , solve $a \circ 2 = 2 - a$ .

The definition gives $a^{2} = 2 - a$ .

So $a^{2} + a - 2 = 0$ , which factorises as $(a + 2) (a - 1) = 0$ .

Therefore $a = - 2$ or $a = 1$ .

Example 18: Suppose $m * n = mn + m + n$ . Find the identity element.

Let the identity be $e$ . Require $m * e = m$ :

$m e + m + e = m$ .

So $e (m + 1) = 0$ for every allowed $m$ , which gives $e = 0$ .

Check the other order: $0 * m = 0 + m + 0 = m$ . Thus the identity is 0.

Example 19: Suppose $m * n = mn /2$ and the identity is 2. Find the inverse of $- 5$ .

Let the inverse be $x$ . We need $(- 5) * x = 2$ .

So $- 5 x /2 = 2$ , giving $- 5 x = 4$ and $x = - 4/5$ .

Example 20: Let $x * y = 3 x + 2 y - 1$ . Is this operation commutative?

$x * y = 3 x + 2 y - 1$ , while $y * x = 3 y + 2 x - 1$ .

These are not equal for all $x, y$ . For instance, $1 * 2 = 6$ but $2 * 1 = 7$ .

Therefore the operation is not commutative.

A simple exam plan

For ordinary sets, write the elements explicitly when that makes the operation easier to see.
Never repeat an element in a set answer.
For complements, identify the universal set first.
In Venn counting, put the intersection in first and use inclusion-exclusion carefully.
Do not assume everyone belongs to at least one set unless the question actually says so.
For a custom binary symbol, ignore its appearance and use only the stated rule.
For nested binary operations, evaluate the innermost operation first.
For identity or inverse questions, translate the definition into an equation and solve it.

Check that your final answer is consistent with the original sets or operation before moving on.

Sets and Binary Operations

Set Concepts & Operations

Venn Diagrams & Set Counting

Binary Operations

A simple exam plan

Key terms

Practise Sets and Binary Operations questions

Related topics

More JAMB Mathematics notes