Relational algebra formulas

Every superset of a superkey (within the attributes of $R$) is a superkey, so under set semantics the set of all attributes is always a superkey. Candidate keys are the minimal superkeys.

$n$ is the number of attributes of $R$. A superkey is any attribute set that contains at least one candidate key. With two keys, inclusion-exclusion subtracts the sets counted twice: those containing $K_1\cup K_2$.

Keeps whole tuples satisfying the condition; schema unchanged; $|{\sigma }_{\theta }(R)|\le |R|$.

Keeps the listed columns and removes duplicate rows (set semantics). For non-empty $R$, $1\le |{\pi }_L(R)|\le |R|$, with $|{\pi }_L(R)|=|R|$ whenever the list $L$ contains a superkey.

Pairs every tuple of $R$ with every tuple of $S$. Clashing attribute names are qualified ($R.A$, $S.A$) or renamed with $\rho$.

Defined only for union-compatible relations: the same degree, with the $i$-th attributes of both having the same domain. By convention the result takes its attribute names from the first operand. Union is commutative and associative; difference is neither.

Set semantics, $R$ and $S$ union-compatible. Also $|R\cup S|=|R|+|S|-|R\cap S|$ and $|R-S|=|R|-|R\cap S|$. The upper bounds of $\cup$ and $-$ are reached when $R\cap S=\varnothing$; their lower bounds when the smaller relation is contained in the larger.

$0\le |R{\bowtie }_{\theta }S|\le |R|\cdot |S|$. An equi-join is a theta join whose condition is only equalities; it keeps both copies of the compared attributes.

$C_1,\dots ,C_m$ are the attributes with the same name in both relations, and the result has degree $\mathrm{deg}R+\mathrm{deg}S-m$. With no common attribute it equals $R\times S$; when the two schemas are identical it equals $R\cap S$. A NULL never matches, not even another NULL. Natural join is commutative and associative (up to column order).

Union-compatible relations; $|R\cap S|\le \min (|R|,|S|)$.

$R(X,Y)$ and $S(Y)$, where the attributes $Y$ of $S$ are a proper subset of those of $R$ and $X$ is the rest. The result has schema $X$ (degree $\mathrm{deg}R-\mathrm{deg}S$): the $X$-values that appear in $R$ paired with every tuple of $S$. If $S$ is empty, the formula gives $R÷S={\pi }_X(R)$.

Each $X$-value in the result needs its own $|S|$ tuples in $R$, one for each tuple of $S$, and these tuples are different for different $X$-values.

$|R\bowtie S|$ is the inner (natural) join; $u_R$ and $u_S$ are the numbers of tuples of $R$ and of $S$ that match no tuple of the other relation. Left outer adds only $u_R$, right outer only $u_S$.

If the common attributes form a key of $S$, each tuple of $R$ matches at most one tuple of $S$, so $|R\bowtie S|\le |R|$. If they are also a foreign key of $R$ referencing $S$ with no NULL in it, the size is exactly $|R|$.

${\rho }_T(R)$ is a renamed copy of $R$. The subtracted part is every $A$-value that some other value of $A$ exceeds, so what remains is the maximum value of $A$. Use $>$ in place of $<$ for the minimum. This assumes $A$ contains no NULL.

The result is the set of all tuples $t$ for which formula $P$ is true. $t$ is the only free variable of $P$; every other tuple variable must be bound by a quantifier.