Draw a diagraph for the relation R, where (x, y) ER if |x +y z, y E-4, -3,-2,-1,0,1,2,3, 4 2, for
Which statement is false? o 3 x E R and SY ER such that (a +y=1) 1 (x - y = 5). O x E R and Sye R such that (x + y = 1) ^ (3x + 3y = 4). O V x ER, SY ER such that x+y=1. O 3 x E Z and 3 y E Z such that (x+y=1) 1 (x – y = 5).
Define a relation R from R to R as follows: For all (x, y) E R x R, (x, y) E R if, and only if, x= y2 + 1. (a) Is (2, 5) E R? Is (5, 2) e R? Is (-3) R 10? Is 10 R (-3)? (b) Draw the graph of R in the Cartesian plane. (C) Is R a function from R to R? Explain.
(e) Define a relation R on Z as xRy if and only if m|(x - y). Prove that R is an equiv- alence relation.
11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
For r ∈R, let Ar = {(x,y,z ∈R^3 . . . x^2 + y^2 −z^2 = r}. Is this a partition of R^3? If so, give a geometric description of the partitioning sets (i.e., the equivalence classes of the induced equivalence relation).
Consider the sequence of functions fn3. where 'x ER. (a) For which z e R does the series of functions (x) converge pointwise? -1
Q 4. Confirm that ∇ (1/ r) = − r /r 3 where r = ||r e || and r e = xˆı + y ˆj + z kˆ = ρ eˆρ + z eˆz = r eˆr. Do it in (i) cartesian coordinates with ∇ ≡ ∂ ∂x ˆı + ∂ ∂y ˆ + ∂ ∂z kˆ. (ii) cylindrical coordinates with ∇ ≡ ∂ ∂ρ eˆρ + 1 ρ ∂ ∂φ eˆφ + ∂ ∂z eˆz. (iii) spherical coordinates...
Evaluate ∫∫∫ E √ x 2 + y 2 + z 2 d V where E lies above the cone z = √ x 2 + y 2 and between the spheres x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = 9 . df (76 KB) 2. Evaluate r2 + y2 + 22 dV x2 + y2 and between the spheres r? + y2 + 2 = 1 and...
Given the following binary relations: The relation Rl on {w, 1, y, z), where R1 = {(w, w), (w, 1), (x, w), (x, 1 ), (x, z), (y, y), (z,y),(2, 2)). The relation R2 on (a, b, c), where R2 = {(a, a ), (b, b), (c, c), (a, b), (a, c), (c, b)}. The relation R3 on {x,y,z}, where R3 = {(1, 2), (9,2), (2, y)}. Determine whether these relations are: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive?
Consider the joint density function fX,Y,Z(x,y,z)=(x+y)e−zfX,Y,Z(x,y,z)=(x+y)e−z where 0<x<1,0<y<1,z>0. b) Find the marginal density of (x,z) : fX,Z(x,z). For your spot check, please report fX,Z(1/2,1/4)+fX,Z(1/4,1/2)+fX,Z(1/2,2) rounded to 3 decimal places.