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: For the relation R closure of R. {(1,2), (1,4), (3,3), (4, 1)} find (i) R2 (ii) R3 and (iii)…

A: The given relation,

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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…

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Q: 2. The relations below are defined on the set {0,1,2,3}. Determine which of these relations are…

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

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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…

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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-…

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Q: Suppose that R is a symmetric relation. Match the correct term with each blank space in the…

A: Suppose  that R is a symmetric relation than prove that R-1⊆R Proof: Suppose that &lt;x,y&gt;∈R-1.…

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: 3. For universe S = R² define relation ~ as: (a, b) ~ (c,d) ⇒ 2a-b2c - d. Prove that is an…

A:

Q: 8. (6 points) Define a relation R on Z as follows: For all integers m and n, m R n ⇒ (m − n)is a…

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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.