Define relations S and R on the set A = {x, y, z, w} as follows S = {(y, x), (x, w), (w, z), (x, x)}, R%3D (х, у), (w, и), (у, w), (z, х)}. then S. S is O{(y,w), (y,x), (x,z), (x,w), (x,x)}. OL{(y,w), (x,z)}. O{(y,w), (y,x), (x,z), (x,w)}. O{(y,w), (y,x), (x,z)}. Onone of the other answers. and S• R•S-l is O{(x,z), (w,x), (z,z),(x,x)}. O{(x,y), (w,w), (y,w), (z,x) }. Ol{(x,y), (w,w), (w,y), (z,x), (x,w)}. O{(x,2), (w,x), (z,z)}. Onone of the other answers. Pick the correct statement among the following: OR is a function but S and S R are not functions. OR and S o R are functions but S is not a function. OAII R, S and S• R are functions. OAII R, S and S R are not functions. ONone of the other statements is correct.

Q: Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. (i) Prove that…

A: Given that X,Y,Z are subsets of {1,2,3,...,10} |X|=|Y|=|Z|=7. X∪Y≤10 i)X∩Y≥4 X∪Y=X+Y-X∩YX∩Y=X+Y-X∪Y…

Q: 5. Define a relation S from R to R as follows: For all (x,y) ERx R, (x,y) €S means that x > y. a. Is…

A:

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

A:

Q: Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {6, X, {a}, {a,…

A: I

Q: Let f = 12r + kry + Jryz re-express f O f(r,y,z) = (x, ry, Tyz+) O {r,y) = (x, ry, ryz*) O f(x,y) =…

A: GIVEN f= ix+kxy+jxyz4     ....(1)

Q: Prove that if w=f(x, y, z), x=r-s, y=s-t, and z=t-r then dw dw dw + + = 0. dr ds ôt

A:

Q: Consider a bounded function f [a, b] → R and a partition P = U(f, P) ≥ U(f, Q) and L(f, P) ≤ L(f,…

A:

Q: Vx E X, (x, x) ¢ R. X = {1,2,3}, h of the following relations R on X is irreflexive?

A: Here given set   X={1,2,3}   we know that relation is irreflexive when each x belong X (x,x) dose…

Q: Let Z = {0, 1, 2, 3} and the relations R, S, and T on Z are as follows:

A:

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

A:

Q: find g(u(x,y,z), v(x,y,z), w(x,y,z)) if g(x,y,z)=zsinxy, u(x,y,z)=x2z3, v(x,y,z)=πxyz, and…

A: Given, g(x,y,z)=zsinxyu(x,y,z)=x2z3v(x,y,z)=πxyzw(x,y,z)=xyz     Find g(u(x,y,z),v(x,y,z),w(x,y,z)):

Q: If G (x, y, z) = curlF (x, y, z), where - F(x, y, z) = (y — z, z — x, x - y), then div(curlG (x, y,…

A:

Q: (a) Prove the following relation: B(x + 1, y): B(x,y) x + y %3D

A:

Q: Let P, be an equidistant partition of [2, 4] on f(x) = x , then U(f, P,) =

A: To find the U(f, pn) of the function f(x) = x

Q: Let F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

A:

Q: Show that if Sı and S2 are convex sets in R"+n, then so is their partial sum S = {(x, yı + y2) | x €…

A: We will solve the question number 2.16 as mentioned. Given that S1,S2 are two convex sets in ℝn+m.…

Q: Let W = (0, 1, 3}, X = {1, 3, 5), Y = (3, 4} and Z = {(1, 2), (3, 4), (1, 3),(2, 4). Compute (X x Y)…

A: The given problem is to find the Cartesian product of given sets and verify the Cartesian product…

Q: Let W(s,t) F(u(s, t), v(s, t)) where u(1,0)-4, u,(1,0) 3, u(1,0)- 5 v(1,0) 7, v.(1,0)=-3, v(1,0)--6…

A:

Q: The set S = {T, (U) | U is open in X} U {T, (V ) | V is open in Y } is a subbasis for the product…

A: The given theorem is related with the topic product topology. Given that S = π1-1U | U is open in X…

Q: Let X1,., X, N(µ, o²). Find iid the bias of i = X3 +X + X, if u

A: Note : Since you have asked multiple questions, we will solve the first question for you. If you…

Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: Suppose z = z(x,y) is x + y+z = e¬(x+y+z) , then which relation is correct? az D): az az az az 1 ду…

A: The solution is given as follows :

Q: if u=f(x,y), x=g(w) and y=h(w), using chain rule, what is ∂u/∂w?

A: Silution : Given that u = f(x,y)

Q: Let x, y € R" be given. (a) If ||z||||y||=1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1 and ||y||…

A:

Q: Let r = ⟨x, y, z⟩ and r0 = ⟨1, 1, 1⟩. Describe the set of all points (x, y, z) such that r − r0 =…

A: Given: r=x,y,z and r0=1,1,1 To describe the set of all points x,y,z such that r-r0=2 Solution:…

Q: If G (x, y, z) = curlF (x, y, z), where F (x, y, z) = (y – z, z – x, x – y), then div(curlG (x, y,…

A: given statement if false.

Q: Prove: If U is a neighborhood of x0 and U ⊂ V, then V is a neighborhood of x0. Also, if U1 ..., Un…

A: The neighborhood of a point: Let (X, T) be a topological space and x0∈X. A neighborhood of x0 at is…

Q: In Exercises 1–3, begin by drawing a diagram that shows the relations among the variables. 1. If w =…

A:

Q: The domain of the function f(r, y, z) = In(6 2xy) is O {{z, y, z) E R³ ° : ay > 3} O {{z, y, 2) E R°…

A: see below the answer

Q: A relation R on a set A is called anti-symmetric if O (a, b) E R implies (b, a) E R for every a, b e…

A:

Q: (x), we - Let (-) be the evaluation inner product on P₂ at -1, 0, and 1. That is, for any p = p(x)…

A:

Q: Let F(x, y, 2) = (z, 1, y). a. Find divF and curlF.

A:

Q: Let R be the ternary relation on Z that is defined as R:= {(x,y, z) E Z × Z × Z : a² + y² + 2² < 5}…

A: If a relation is defined on Z, then the variables in the relation can have only integer values to…

Q: (a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by…

A:

Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by f X,Y (x, y) = (x+y)/21…

A: Solution: The joint pmf of X and Y is  fX,Y(x,y)=(x+y)21     x=1,2,3    and y= 1,2  The pmf of X and…

Q: If G (x, y, z) = curlF (x, y, z), where F (x, y, z) = (y– z, z – x, x – y), then div(curlG (x, y,…

A:

Q: Let X, Y ~ U[a, b] be independent. Define Z = max{X, Y}. Find the cdf of Z.

A: Let X, Y ~ U[a, b] be independent.

Q: Find det (T) for T:R- R, where T(x,,×2,*3) = (x,- x2, X2 - X3, X3-x).

A: We will find the matrix [T] from the image of the standard basis vectors of R³ under the…

Q: Let f(x, y, z) = xyt + z3 and x = st², y = s³t² , and

A: The given function is fx,y,z=xy4+z3 and the values of x, y, and z are st2, s3t2, and s2t2…

Q: Determine which of these relations are transitive. The variables x, y, x', y' represent integers.…

A:

Q: Is anyone of the veetors e. 2-, ERa linear combination of the others? If so, find the relation(s)…

A:

Q: The domain of the function f (x, y, z) = In(6 – 3ry) is - o {(x, y) E R² : xy > 2} O {(x, y, z) E…

A: Please refer to the image below

Q: Assume that z = ƒ(w), w = g(x, y), x = 2r^3 - s2, and y = re^s. If g x(2, 1) = -3, g y(2, 1) = 2,…

A:

Q: Let W (s, t) = F(u(s, t), v(s, t)) where W,(1,0) WŁ(1,0) = u(1,0) = -1, u,(1, 0) = −5, ut(1,0) = -7…

A:

Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A: Let, s be the circle given by, s=x,y∈ℝ2 :x2+y2=2  Then s is a closed subset of ℝ2. Let us consider a…

Q: Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b, c, d}, Tx = {ø, X, {a}, {a,…

A: We need to check the necessary and sufficient conditions for continuity

Q: Find f and f, and evaluate each at the given point. y' f(x, y) = e*y, (In 4, 2) f,(x, y) = f,x, y) =…

A:

Q: Let f(x, y, z) = 2x(y+z), x uv, y = v w, and z = w. Find in terms of u, v, and uw.

A:

Q: Prove that if w = f(x,y,z), x=r-s, y=s-t, and z =t-r then ôw ôw ôw + âr' as = 0. ôt

A:

Q: Let (X, Tx), (Y, Ty) represents topological spaces such that X = {a,b,c,d}, Tx = {X. Ø, {a}, {a, b},…

A: CONTINUOUS FUNCTION: If the inverse image of every open subset of Y is open in X, then the function…

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.