✓2.39 Recall that the null space is a subset of the domain and the range space is a subset of the codomain. Are they necessarily distinct? Is there a homomorphism that has a nontrivial intersection of its null space and its range space?

Prove that if f is a permutation on A, then (f1)1=f.

Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.

29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there is a subset such that . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets .

II. Let Q-set of all real numbers, A-index set. Identify the generalized union and intersection of the following: 1. A₁ [0,4], A2-[1,5), A3- [0.5,3], A4- (1,9] As-[-5,8) where A = {1,2,3,4,5} 2. A [0,4], A₂-[1,5), A3- [0.5,3], A=(1,9] As-[-5,-2) where A = {1,2,3,4,5}

26 of 40 Consider the Datalog programs P1 (left) and P2 (right) below, which use relations R(A, B) and S(A, B). P1 P2: T1(A) R(A, B). T4(A) R(A, B), S(A, B). T2(A) S(A, B). T3(A) + T1(A), T2(A). Which of the following statements is TRUE about the relationships between relations T3 and T4 defined by P1 and P2, respectively? Note that the commas "," used in the rule bodies to separate the predicates is the same as using AND. Select one: T3 and T4 include the same set of tuples. Every tuple in T3 is also contained in T4, that is, T3 C T4. O None of the other answers, that is, T3 and T4 contain different tuples, in general. O Every tuple in T4 is also contained in T3, that is, T4 C T3.

What is the kernel and image of the map

Let Ω-set of all real numbers, A-index set. Identify the generalized union and intersection of the following:

If As ş a b.} which of these relations are a funchion? Exploin.

Let f(r) = Vt3 +1 dt for r2 3. 3