Q5. Let X = {a, b, c, d, e, f} and 7 be the discrete topology on X. Which of the following statements are true? 1) X ET;, 2) {X} ET;, 3) {0} ET;, 4) 0 ET; 7) {a} ET, 8) a Є T;, 9) 0 CX;, 10) {a} = X;, 5) 0 Є X;, 6) {0} = X;, 11) {0} CX;, 12) a Є X;, 1 13) XCT;, 14) {a} CT;, 15) {X} CT;, 16) a CT.

Prove or disprove that AB=AC implies B=C.

Let all of the lines named be coplanar. Make a drawing to reach a conclusion. a If rs and st, then _. b If ab and bc, then _.

10. Let A, B, and C be sets. Use the identities to show that (A U B) n (B U C) n (A U C) = Ā n Ēn T.

Let be 7= {x, , {a}, {cd}, {a,c,d}; {b, c, d,e}} topology on X= {9₁b, c, d, e}. Check if X is connected or not.

19. Using the Venn diagram below, identify the elements for the indicated set. U 2 6 COD: 5 3 9 7 10 a) X d) XUY 1 8 4 c) U f) Y - X b) Y e)XnY g) XUY h) Xn Y 20. In the spaces provided, place the symbol that will make the relationship true.

B. Determine whether the following sets are equivalent, equal, or disjoint sets.  1.) B = {red, yellow, blue} and D = {blue, red, yellow} 2.) A = {1, 2, 6, 7, 10} and F = {v, w, x, y, z} * 3.) C = {a, b, c} and E = {w, x, y} 4.) D = {5, 10, 15, 20} and H = {7, 14, 21, 28, 35, 49}  5.) B = {banana, apple, mango, grapes} and D = {1, 2, 3, 4} *

2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a, b) : a € A and be B}. This can be extended naturally to a finite number of sets A1, A2, … , Aµ: A1 x Az x -..x Ak = {(a1, a2, ...,ak) : a; € Aj for 1< j< k}. %3D This can be further extended to an infinite number of sets. Let I be an index set and X = {X;}ie1 be a family of sets indexed by the set I. Note that I may be an uncountable set. The Cartesian product of the family X is given by II x X, = {s :1¬UX; : f6) e X; for all i e I iel That is, the elements of the Cartesian product are functions whose domain is the index set I and the image of i is in X; for any i € I. Let {G¡}ie1 be a family of groups indexed by the set I. Let e; be the identity element of G¡, for all i e I. Consider the Cartesian product G = IIies Gi- %3D a. Show that G is a group under multiplication given by fg(i) = f(i)g(i) for all e I. We call G the direct %3D product of the family {G;}ie1-

pls. answer correctly... thank you

2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a, b) : a € A and b e B}. This can be extended naturally to a finite number of sets A1, A2,…., Aµ: Aj x Az x ...x Ag = {(a1, a2, …..,ak) : a; € A; for 1< j< k}. This can be further extended to an infinite number of sets. Let I be an index set and X = {X;}ie1 be a family of sets indexed by the set I. Note that I may be an uncountable set. The Cartesian product of the family X is given by %3D Įx = {s:1 →UX : f(1) € x,for ll 1 i! That is, the elements of the Cartesian product are functions whose domain is the index set I and the image of i is in X¡ for any i e I. Let {G¡}ie1 be a family of groups indexed by the set I. Let e; be the identity element of G¡, for all i e I. Consider the Cartesian product G = [IIkes Gi- c. Let {p;}1 be the set of all the prime numbers. Show that any rational number q E Q can be written in the form q = E where n; € Z+ for all j € Z+, and all but finitely many of the x;'s are…