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

10. Prove that for subsets X, Y, and Z of a given universal set U (a) X-Y=X-(XY)= (XUY) -Y (b) (X-Y)-Z=X - (YUZ) (c) X-(Y-Z) = (X-Y)U(Xnz) (d) (XUY) -Z=(X-Z) U(Y - Z) (e) X-(YUZ) =(x-Y)^(X-Z) Illustrate each case on a Venn diagram.

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

10. (a) Define the independence of sets A, B, C. (b) Provide an example where A, B, C are pairwise independent but not mutually independent. (c) Give an example where P(AnBnC) = P(A)P(B)P(C), but the sets are not pairwise independent.

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}.

Which of the following are true for all sets A, B, and C. If the statement is true, prove it by showing each is a subset of the other. If not, provide a counterexample. (d) (A')' = A(e) A \ B = (B \ A)'(f) (A \ B) ∩ (B \ A) = ∅(g) If A ∩ B = ∅, then A ⊆ B.

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

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

1. Let 1 -0-0-0-0 = = = (a) (5 points) Find a orthonormal set {q1, 92, 93, 94} such that span{a₁, ..., a¿} for i = 1,2,3,4. Give exact answer and show your workings clearly. (b) (5 points) Let A = = span{1,..., q} QR = (a₁ a2 a3 a4) and Q = (91 92 93 94). Write A for some upper triangular matrix R. Give exact answer and explain how you derive your answer. (c) (2 points) Find the projection of workings clearly. 3 onto V = span{q1, 92, 93, 94}. Show your

Let the sets A, B and C be fuzzy sets defined on real numbers by the membership functions: HA(x)= x/(x+1), Ha(x)= 1/(x²+10), µc(x)= 1/10*. Determine mathematical membership functions and graphs of each of the following: 1. AUB X' 2. BnC 3. AUBUC 4. ANBNC 5. An Complement(C) 6. Complement(B) U C 7. Complement(A O B) 8. Complement(A) U Complement(B).