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Put $J=\{k\in \mathbb{N}:k\le 100\}$, avoiding any dependence on whether the convention includes $0$ among the natural numbers. Under set inclusion, the results are

The set union of all three-element subsets of $\mathbb{N}$ is $\mathbb{N}$. An upper bound for $S_1$ under set inclusion would have to contain this entire infinite set, whereas members of $A$ are finite. Thus $S_1$ has no upper bound and hence no maximum or least upper bound.

For $S_2$, every member is a subset of the finite set $J$, and $J$ itself belongs to $S_2$, making it both maximum and least upper bound. The union of the three-element subsets of $J$ is again $J$, so $J$ is the least upper bound of $S_1\cap S_2$. It is not a member of that family, and no three-element member contains all the others, so there is no maximum.

Put $B=T\cap S$. By distributivity, $g(A)=B\cup (T\cap A)$ and $B\subseteq T$. Consequently,Here we used associativity, distributivity, and idempotence of set intersection and set union. The equality holds for every $A\in \mathcal{P}(U)$, so $g\circ g=g$: $g$ is an idempotent function.

The prime omega function $\omega (n)$ counts the distinct prime factors of a positive integer $n$, so it is the cardinality of the prime support of an integer $P(n)$. For a prime factorization with distinct prime numbers $p_1,\dots ,p_r$ and positive exponents,At $n=1$ there are no prime factors, so the empty sum gives $\omega (1)=0$. The total number of prime factors instead gives $\Omega (n)={\sum }_i{a}_i$. ^[ref]

For $72=2^3\cdot 3^2$, the prime support of an integer is $\{2,3\}$ and $\omega (72)=2$. The comparison examples give $\omega (30)=3$, $\omega (32)=1$ and $\omega (36)=2$. Raising an existing prime factor to a higher power does not affect the prime omega function: $\omega (p^k)=1$ for a prime number $p$ and $k\ge 1$. Thus the prime omega function is a strongly additive arithmetic function, withThe reason is that coprime integers have disjoint prime supports of an integer. For arbitrary positive integers, their prime support of an integer combines by set union, and the shared prime factors are exactly those of the greatest common divisor. Counting this set union gives the more general identityFor $a=12$, $b=18$, both prime supports of an integer are $\{2,3\}$ and $\gcd (a,b)=6$, so $\omega (216)=2+2-2=2$. In particular $\omega (4)=1\ne 2=\omega (2)+\omega (2)$: the prime omega function is not a completely additive arithmetic function.

The prime omega function is useful for counting choices involving distinct prime factors. Every squarefree divisor of $n$ chooses a subset of its prime support of an integer, so there are $2^{\omega (n)}$ squarefree divisors. For $72$, this gives $2^2=4$, namely $1,2,3,6$. Also $\omega (\mathrm{rad}(n))=\omega (n)$, because the radical of an integer keeps exactly that prime support of an integer. The total number of prime factors records what this count discards, and the two counts agree exactly on squarefree integers.

The set difference $A\setminus B$ keeps the set elements of $A$ that do not belong to $B$:Thus set difference combines logical conjunction (AND) with logical negation (NOT): pass the first condition and fail the second. The definition applies to any two sets; $B$ need not be a subset of $A$.

For $A=\{1,2,3\}$ and $B=\{3,4\}$, the two orders give different answers:The common member $3$ is removed in both directions. A database filter illustrates the usefulness: if $A$ is the set of records matching the first condition and $B$ the set matching the second, $A\setminus B$ selects records matching the first but not the second. The Venn diagram in set union shades both directions separately.

The name relative complement emphasizes that the ambient set is $A$: $A\setminus B$ is the complement of $A\cap B$ within $A$. If a larger universal set $U$ contains $A$ and $B$, then $A\setminus B=A\cap (U\setminus B)$. Changing that larger universal set does not change the set difference, because membership still requires belonging to $A$. In ZF the set difference exists by the axiom schema of separation. It also gives relative complements within a fixed tuple domain in finite relation closure for set-theoretic coding.

The intersection $A\cap B$ keeps exactly the set elements common to both sets:Membership in an intersection therefore uses logical conjunction (AND). For $A=\{1,2,3\}$ and $B=\{3,4\}$, the only shared member is $3$, so $A\cap B=\{3\}$. The Venn diagram in set union shades only the overlap for this operation.

An intersection is useful when every requirement must hold. If $A$ and $B$ are the sets of database records passing two separate conditions, their intersection gives the records passing both. Similarly, satisfying two equations requires belonging to the intersection of their solution sets. The same membership rule applies even when the objects have entirely different meanings.

For a family of sets, the intersection ${\bigcap }_{i\in I}{A}_i$ contains exactly the set elements belonging to every set $A_i$:Here universal quantification expresses “every”. If all the sets lie within a fixed universal set $U$, the intersection over no indexing members is defined to be $U$: there are no requirements to fail. This convention depends on specifying $U$. By contrast, intersecting with the empty set always gives $A\cap \varnothing =\varnothing$.

A measure is sigma-finite when its space is a countable union of measurable sets of finite measure.

A Venn diagram represents sets by closed curves, with each region recording a possible pattern of membership. For two sets, the regions mean “first only”, “both”, “second only”, and “neither”; the enclosing rectangle can represent the universal set. Shading regions displays union, intersection, set difference, or complement. In the numbered examples in set union and complement, positions encode membership, while curve shapes and region areas do not encode cardinalities.