Sets and Set Operations

≈ 20 min

A set is a collection of distinct objects called elements. We write $x \in A$ for ' $x$ is in $A$ '. Key operations:

The number of elements in $A$ is its cardinality $|A|$ .

For overlapping sets, simply adding sizes double-counts the shared part. The inclusion–exclusion principle fixes this:

|A \cup B| = |A| + |B| - |A \cap B|

Worked example. If $|A|=10$ , $|B|=8$ and $|A\cap B|=3$ , then $|A\cup B| = 10 + 8 - 3 = 15$ — we subtract the 3 counted twice.

A set with $n$ elements has $2^n$ subsets: each element is either in or out.

If $A=\{1,2,3,4\}$ and $B=\{3,4,5\}$ , what is $A \cap B$ ?

$\{1,2,3,4,5\}$

$\{3,4\}$

$\{5\}$

$\{1,2\}$

If $|A|=10$ , $|B|=8$ and $|A\cap B|=3$ , find $|A\cup B|$ .

How many subsets does a set with 4 elements have?

In a class of 30 learners, 18 take Mathematics and 15 take Physical Sciences; 7 take both. How many take at least one of the two subjects?