In my notes on canonical disjunctive normal form in Boolean algebra I have written:
A monom is a finite conjunction of literals $L_1,...L_m$. The universe $U$ is a degenerate monom. A formula is in disjunctive normal form if it is a finite disjunction of monism $M_1,..., M_n$. The empty set $\emptyset$ is in DNF degenerately.
Why is the universe $U$ a degenerate monom? Why is the empty set in DNF degenerately? What does this mean?
A
Disjunctive Normal Form(DNF) is also called Sum of Products. The products (or monoms) areANDexpressions of literals (inverted or non-inverted inputs). If the sum or disjunction of products does not contain any products, this is an empty set. The expression is not fulfilled for any combination of input values.The opposite case of an empty set of products is a set which contains all $2^n$ products. They constitute a full truth-table. In this case, the expression is fulfilled for every combination of input values. The complete set of all possible products is sometimes called universe. If literals of a product are dropped one-by-one by merging adjacent products with just one different literal, the remaining product value in the degenerate case without any literals left is $1$.