By Tripathi M. M., Kim J., Kim S.

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**Additional info for A basic inequality for submanifolds in locally conformal almost cosymplectic manifolds**

**Example text**

P(0, 1, 0, 78). p(0, 1, 1, 398). p(1, 0, 0, 87). p(1, 0, 1, 398). p(1, 1, 0, 49). 62 Prolog Experiments p(1, 1, 1, 374). For each of the following cases, use setof, bagof, or findall to construct the indicated set or bag. a. The set of all facts of the form p(0, X, Y, Z) for some X, Y, and Z. b. The bag of all numbers N such that p(X, Y, 0, N) for some X and Y. c. The bag of all numbers N such that p(X, Y, Z, N) for some X, Y, and Z. d. The bag of all pairs [Y, Z] such that p(X, Y, Z, N) for some X and N.

We’ll start with the subset relation. Recall that A Ã B if and only if every element of A is an element of B. It follows from this that the empty set is a subset of every set. This gives us the basis case for the following recursive definition of the subset predicate. List Structures 63 subset([ ], _). subset([H|T], X) :- member(H, X), subset(T, X). For example, the following goal returns yes. - subset([a, b], [c, a, d, b]). Now we can use the subset relation as a tool to define a test for equality of sets.

Let’s try to use the setof predicate to find S. ” We can compute the set S with the following goal. - setof(X, Y^p(X, Y), S). This goal returns S = [a, b, c]. For another example, suppose that we want to calculate the set of second arguments that have the letter a as a first argument of p. In other words, we 60 Prolog Experiments want to calculate the set S = {x | p(a, x)} = {b, c}. We can compute the set S with the following goal. - setof(X, p(a, X), S). This goal returns S = [b, c]. For another example, suppose that we want to construct the set of “grandparent” relationships from p.