An object-oriented database programming environment for by Supcik J.

By Supcik J.

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D = {x ∈ N : 2 ≤ x ≤ 13 and x is prime}. Solution 3. The powerset is {∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {b, c, d}, {a, b, c, d}}, and it has 16 elements. 46 S ETS , R ELATIONS , F UNCTIONS Solution 4. We note that • P(∅) has 1 element, • P({a}) has 2 elements, • P({a, b}) has 4 elements, • P({a, b, c}) has 8 elements, • P({a, b, c, d}) has 16 elements. Thus, we conjecture that the powerset of a set with n elements has 2n elements. Solution 5.

Let A be a set of objects each of which has only one colour; these could be blocks or balls or similar things. If we are teaching a child to distinguish colours, we might ask the child to make heaps of those objects which have the same colour. By doing this, we are in fact training the child to work with the relation R on A, which is defined by xRy if x and y have the same colour. A heap of objects then would be the set {y ∈ A : xRy} for some fixed x ∈ A; we would point to one object (our fixed x), say, a blue tile, and would ask the child to collect all blue objects (all y of the same colour).

Clearly, the relation ≡ is reflexive; it is also transitive: Let a ≡ b and b ≡ c; then there are k, n, mN , and r, s, t 2 such that a = 3k + r, b = 3n + s, c = 3m + t. Since a ≡ b, we must have r = s, and since b ≡ c, we have s = t; thus, r = s = t, and we see that a = 3k + r, c = 3m + r, which tells us that a ≡ c. Hence, ≡ is transitive. However, ≡ is not antisymmetric; take for example a = 4, and b = 7; then 4 ≡ 7 and 7 ≡ 4, since they both have remainder 1 when divided by 3. But this example indicates that ≡ has another property: If a ≡ b then b ≡ a, for any a, b ∈ N.

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