(d) (XUY) -Z=(X-Z)U(Y-Z) (e) X(YUZ) = (X-Y)^(X-Z)

1. (a) (b) (c) (d) Prove or disprove that, for any universal set U and predicates P and Q, [3x = U, P(x) ^ Q(x)] → [3r EU, P(x)) ^ (3x = U, Q(x))] Prove or disprove that, for any universal set U and predicates P and Q, [3r EU, P(x)) ^ (3xU, Q(x))] → [r U, P(x) ^ Q(x)] Prove or disprove that, for any universal set U and predicate P [3r € U, P(x)] → [Vr € U, P(x)] Prove or disprove that, for any universal set U and predicate P [VxU, P(x)] → [3r € U, P(x)]

(4) Let A, B, C, D be sets. (a) (b) Prove that Ax (B \ C) = (A × B) \ (A × C). Prove that (A x B) n (C x D) = (ANC) × (BND).

X, Y and Z be subsets of the set . Verify the following properties: XU(YUZ) = (XUY)UZ, Xn(Y_Z) = (XnY)nz (Associativi XU(YNZ) = (XUY)N(XUZ), XN(Yuz)=(XNY)u(XNZ) XcY+XUY=Y+X Y=X

A = { 3, 0, 1 } B = { y, x }  what is the cardinality of A x P(B) ?

There are 16 binary relations on the set {0, 1}: (a) { } (d) {(1,0)} (g) {(0, 0), (1, 0)} (j) {(0, 1), (1, 1)} (m) {(0,0), (0, 1), (1, 1)} (b) {(0, 0)} (e) {(1, 1)} (h) {(0, 0), (1, 1)} (k) {(1, 0), (1, 1)} (c) {(0, 1)} (f) {(0, 0), (0, 1)} (i) {(0, 1), (1,0)} (1) {(0,0), (0, 1), (1, 0)} (0) {(0, 1), (1, 0), (1, 1)} (n) {(0,0), (1,0), (1, 1)} (p) {(0, 0), (0, 1), (1, 0), (1, 1)} For a-f use the letters for each relation above. a. List the reflexive relations on the set {0, 1}. b. List the irreflexive relations on the set {0, 1}. c. List the symmetric relations on the set {0, 1}. d. List the transitive relations on the set {0, 1}. e. List the antisymmetric relations on the set {0, 1}. f. List the asymmetric relations on the set {0, 1}.

Prove that for subset X , Y, and Z of given universal set U

1. Find all regular points of the set {x = R¹ | {x € R¹ | +3(x1+x2+x3)² 2(x₁+x2+x3)³ +3(x1+x2 + x3)² = 1, x²+x²+x² + x₁ = 1}. 2(x1+x2+x3)³

1. (i) (ii) which are not. What does it mean to say that a set ECR2 is closed? [1 Mark] Identify which of the following subsets of R2 are closed and (a) A = [-1, 1] × (1, 3) (b) B = [-1, 1] x {1,3} (c) C = {(1/n², 1/n2) ER2 | n EN} Provide a sketch and a brief explanation to each of your answers. [6 Marks] (iii) Give an example of a closed set which does not have interior points. [3 Marks]

6. Show that R - Q and R - Z are not homeomorphic.