2. For universe S = R² define relation as: (a, b) ≤ (c,d) ⇒ 2a-b≤2c-d. For each of the properties below, find a counterexample or prove that it holds for S: (a) reflexivity, irreflexivity, (b) symmetry, asymmetry, antisymmetry, (c) transitivity, intransitivity, From the above deduce whether the above relation is either of below: (a) equivalence (b) partially ordered set (i.e. poset) (c) strictly partially ordered set (i.e. strict poset)

Q: (b) Let R = {(x, y) E R × R : 1 < ¤ < 4 or x = 6}, and S = {(x, y) E R × R : -1 < y < 1}. i. Find…

A: Solution: The given relations are R=x,y∈ℝ×ℝ : 1≤x≤4 or x=6 and S=x,y∈ℝ×ℝ : -1&lt;y&lt;1

Q: 3. f(x, y) = x² x+y + y²-1 -; (1, 1), (-1, 1).

A:

Q: b. Show that if f: A-> B is a bijection then |A| = |B| %3D c. Show that f:25 25 given by f(A) = S -…

A: Aa per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: R.2. Consider the relation R=- ={x,y):x.yeR&]x=}<3}. Is R an equivalence relation on set of real…

A:

Q: c) Let the set A = {-1,0,1} and consider the following function: ƒ3 : A × A → A × A given by f(x, y)…

A:

Q: You are given two graphs G and H with V(G) = {a, b, c, d} and V(H) = {1, 2, 3}. 1. How many…

A:

Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: 4. For (1, yı) and (r2,y2) in R² we set (11. yı) ~ (12. Y2) →{+ =3+ yi. (a) Show that ~ defines an…

A:

Q: Given R = { (x,y) ϵ R | x^2 = y^2, x,y ϵ A } on A = {1, 2, 3, 4, 5}, which of the following…

A:

Q: (b) Let R= {(x,y) E R × R : 1 < x < 4 or x = 6}, %3D and S = {(x, y) E R × R: –1< y < 1}. i. Find…

A:

Q: For universe S = R2 define relation as: (a, b) ≤ (c,d) ⇒ 2a-b≤2c- d. For each of the properties…

A:

Q: Please find a relation R that makes the statement true and a relation R(x, y) that makes the…

A:

Q: IS a function.. (b) Let R= {(x,y) € R × R : 1 < ¤ < 4 or x = 6}, %3D and S = {(x, y) E R × R : –1 <…

A:

Q: 9. For each of the following relations on N, determine which of the five properties are satisfied.…

A: We have to determine properties based on the given relations on ℕ

Q: Determine whether the relation R on the set Z is reflexive, symmetric, antisym- metric, and/or…

A: Given: The relation ℝ on the set ℤ and x,y∈ℝWe need to check whether the given ℝ on the set ℤ is…

Q: R= {(0,2). (1, 1). (1,3), (2, 1). (2, 2). (2,3 For relation R find the Reflexive closure; Symmetric…

A: Given:- Set A={0,1,2,3}   , relation R:A×A is defined as : R={…

Q: Let X = {x € Z+ | 15 < 4x < 22} and Y = {y is even | 0 < 2y + 1 < 15}. Define a relation R from X to…

A: A relation defines the relationship between sets of values of ordered Paris. The set of elements in…

Q: What is the domain of g(x,y) =; ху Select the correct choice below and fill in any answer boxes…

A: Given function is : g(x, y)= 1/(xy)  Domain of function:  The domain of a function is the set of all…

Q: c) Let the set A = {-1,0,1} and consider the following function: f3 A × A → A × A given by f(x, y) =…

A: A function f is said to be injective or one-one if the function value corresponds to only one point…

Q: 1. Let R be the relation on N defined by "aRb if EN or 5| (b- a)". So for example (6, 2) € R and…

A:

Q: A relation R on a set A is called irreflexive iff (Vx E A)((x, x) ¢ R). It is called asymmetric iff…

A:

Q: b) The relation R on A = {a, b, c, d} is defined by the following digraph. b (i) List all paths of…

A: (i) Paths of length 2 are a-b-d, c-d-c, a-c-d, d-d-c, c-c-d, d-d-a

Q: 2. For universe S = R2 define relation as: (a, b) ≤ (c,d) 2a-b≤2c- d. = For each of the properties…

A:

Q: Which of the following relations are functions of y in terms of x? 1) 2) 3) 4) 5- 5+ 5+ 4- 3- 2- 2-…

A:

Q: Find the value of b so that the point x = (13, b) belongs to the boundary of the set U = {x=R² : x₁…

A:

Q: Let fi : R" –→ R and f2 : R" → R be two functions. (5.1) Consider the following proposition: If…

A: let f1:ℝn→ℝ and  f1:ℝn→ℝ be two functions consider the following propositions: if epif1∩epif2 is a…

Q: R.5. Define a relation < on the set of ordered pairs of real numbers (x, y) as follows: (X:V1)< (x2,…

A: The relation ≺ is defined as follows:  x1,y1≺x2,y2 if and only if x1&lt;x2 or x1=x2 and y1≤y2

Q: Consider the relation R = {(x,x): x € Z} on Z. Is R reflexive? Symmetric? Transi- tive? If a…

A: We have to check given relation R for reflexive,symmetric and transitive. Concept required: R:A→B is…

Q: 7. Given set X = {2, 3, 6, 12, 24, 36}. The relation x< is defined as follows: (V x,y E X) x<y→x is…

A: We solve the given problem by using the relation properties and hasse diagram methodology.

Q: 1. Consider the following three domains in the space: {(x,y,z) E R³:1 ≤|x+y+z<4}, {(x,y,z) € R³ : 2…

A:

Q: 3. Let C= {1,2,3} Define relation O from C to C as follows For all (x.y) € Cx C, (x.y) E O means…

A:

Q: Consider the mapping diagram of the relation R below. 2. 3 a) State a counterexample from the…

A:

Q: Let R = {(0, 0), (1, 1), (-1, -1)} be a relation on set A. Deduce the rule between x and y for order…

A: