Disjunction

By Robert Laing

In computer programing, we meet disjunction in various forms: as the or separator between boolean tests, as set union, as database table outer joins, as sum in 2 value algebra, as as parallel statements in flow control.

Or

By Robert Laing

Once again, lets use Stanford University dean Jennifer Widom’s basic SQL examples from her college applications data as an illustration, modifying the same SQL examples done in conjunction as “which students applied for ‘CS’ and ‘EE’?”.

Which students applied for ‘CS’ or ‘EE’?

As with conjunction’s p ∧ q being a spiky version of set theory’s P ∩ Q, disjunction’s p ∨ q is a spiky version of set theory’s P ∪ Q.

Union

Outer Join

By Robert Laing

As explained in inner join relational algebra uses various bowtie symbols to describe binary operations on two tables, which are anologous to combining two sets as I’ll try to explain in this diagram.

The P ∩ Q portion could be considered inner join P ⋈ Q. A left outer join P ⟕ Q is (P - Q) ∪ (P ∩ Q), a right outer join P ⟖ Q is (P ∩ Q) ∪ (Q - P), and a full outer join P ⟗ Q is (P - Q) ∪ (P ∩ Q) ∪ (Q - P).

Sum

By Robert Laing

In 1863 William Stanley Jevons wrote to George Boole that surely Boole’s operation of addition should be replaced by the more natural ‘inclusive or’ (or ‘union’), leading to the law X+X=X. Boole completely rejected this suggestion (it would have destroyed his system based on ordinary algebra) and broke off the correspondence. — The Algebra of Logic Tradition

As can be seen from the above quote, that 1 + 1 = 1 in logic arithmetic caused rancour between the field’s founding fathers, and does to this day. Jevons used a rotated ÷ symbol, ⋅∣⋅, for or instead of + to sidestep the problem.

Parallel