Indicate the properties of each of the following relations. For each relation, indicate whether it is reflexive, anti-reflexive, symmetric, anti-symmetric, and/or transitive. You do not need to justify your answer. 1. R is a relation on set Z such that (x, y) E R if and only if x + y = 0. 2. R is a relation on set Z such that (x, y) ER if and only if x and y have opposite parity. Hint: Two integers have opposite parity if one is even and the other one is odd.

The division operator of relational algebra, “÷”, is defined as follows. Let r(R)and s(S) be relations, and let S ⊆ R; that is, every attribute of schema S isalso in schema R. Given a tuple t, let t[S] denote the projection of tuple t onthe attributes in S. Then r ÷ s is a relation on schema R − S (that is, on theschema containing all attributes of schema R that are not in schema S). A tuplet is in r ÷ s if and only if both of two conditions hold:• t is in ΠR−S(r)• For every tuple ts in s, there is a tuple tr in r satisfying both of the following:a. tr[S] = ts[S]b. tr[R − S] = tGiven the above definition:a. Write a relational algebra expression using the division operator to findthe IDs of all students who have taken all Comp. Sci. courses. (Hint:project takes to just ID and course id, and generate the set of all Comp.Sci. course ids using a select expression, before doing the division.)b. Show how to write the above query in relational algebra, without usingdivision. (By doing so, you…

Select all binary relation properties that apply to the following relation: Equal magnitude: |x| = lyl O antisymmetric O reflexive O symmetric O transitive

21. of 40 A decomposition is in 3NF if every decomposed relation is in 3NF. A decomposition is in BCNF if every decomposed relation is in BCNF. Let R(A,B,C,D,E,F,G) be a relation with the functional dependencies: C→ E, EG→ D, DG-→A, and A→ BF. If we decompose R into relations R1(C, D, G), R2(A, D, G), R3(A, B, F), and R4(C, E), which of the following statements is TRUE? Select one: O The decomposition is in BCNF but Not in 3NF. The decomposition is in 3NF and in BCNF. The decomposition is in 3NF but Not in BCNF. O The decomposition is Not in 3NF and Not in BCNF.