Two sets are equivalent if they contain the same number of elements. The symbol for equivalence is "=" with a bar underneath (though at this level we simply say they are equivalent).

If $n(A) = n(B)$, then sets A and B are equivalent.

Example 1

Show that A = {goat, cow} and B = {2, 4} are equivalent.

Solution

• $n(A) = 2$ (goat and cow)
• $n(B) = 2$ (2 and 4)

Since $n(A) = n(B) = 2$, the sets are equivalent.

Two sets A and B are equal if they contain exactly the same elements, regardless of the order. All equal sets are equivalent, but not all equivalent sets are equal.

Example 2

If A = {1, 2, 3, 4} and B = {4, 3, 2, 1}, are A and B equal?

Solution

Yes. Even though elements are written in different orders, both sets contain the elements 1, 2, 3, and 4. Therefore, A = B.

Set A is a subset of set B if every element of A is also an element of B. We write this as $A \subset B$ or $A \subseteq B$.

• Proper subset ($A \subset B$): All elements of A are in B, but A has fewer elements than B.
• Improper subset ($A \subseteq B$): A is a subset of B and the sets may be equal.

Important facts:

• The empty set ∅ is a subset of every set.
• Every set is a subset of itself.

Example 3

Given A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g}, identify the relationship.

Solution

All elements of A (a, b, c, d, e) are also in B. Since A has fewer elements, A is a proper subset of B:

$A \subset B$

Example 4

List all subsets of A = {a, b}.

Solution

The subsets are: ∅, {a}, {b}, and {a, b}

So there are 4 subsets. The number of subsets of a set with n elements is $2^n$. For A = {a, b}, n = 2, so $2^2 = 4$ subsets.

A universal set (denoted by μ or U) is the set that contains all the elements under consideration. All other sets being discussed are subsets of this universal set.

Example 5

If A = {2, 4, 6} and B = {1, 3, 5}, what could be a universal set?

Solution

A suitable universal set could be μ = {1, 2, 3, 4, 5, 6} — it contains all elements that appear in both sets.

In everyday life, comparing sets helps with organisation and decision-making. For example, a shopkeeper in Dar es Salaam comparing stock sets: if Set A = {items sold today} and Set B = {all items in the shop}, the shopkeeper can determine which items are running low (subsets) or check whether today's sales covered all product categories (equivalent/equal sets). This helps with restocking and budgeting using Tanzanian shillings.