Every function is a relation but every relation is not a(i) Closure property We say that * on S satisfies the closure9: B→ C, thenB anu6. Binary operation Let S be a non-empty set điu* ht ali Operation(y,xIf (X,Yf(x)gof : A C such that:(gof) (x) = g{f(x)},▼ x e AgofsymmeNotefunction.For exFor ton S such thatthenFor ea eS, b eS= a *b e1SVa,b eS.Then, * is called a binary operation on S.3x1oteHere, Aproperty, ifa e S, b eS= a* beS, Va, b eS.a) ForRi(ii) Commutative law OperationS is said to becommutative, ifa* b = b * a V a,b eS.on(iii) Associative law Operation * on S is said to be associative,if (a * b)* c = a * (b * c); a, b, c eS.(© Forz =pa(2,Exercise 1.1(iii) (a) FoQuestion 1.reflexive, symmetric and transitive :(i) Relation RR = {(x, y): 3x-y 0}(ii) RelationR = {(x, y): y = x + 5 and x < 4}(iii) Relation R in the set A={1, 2, 3, 4, 5, 6} as R = {(x, y): y is divisibleby x}(iv) Relation R in the set Z of all integers defined as R = {(x, y): x -y is aninteger }Determine whether each of the following relations arein(b) Fthe set A= {1, 2, 3, ......,13, 14 } definedashonof natural(c)R.inset Nnumbersdefinedas(iv) (aob olnw awREDMI NOTE 5 PROMI DUAL CAMERA

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Exercise 1.1 Question 1. Determine whether each of the following ręlations are eflexive, symmetric and transitive : (i) Relation R = {(x, y): 3x-y 0} R in the set A= {1, 2, 3, ...13, 14} defined as ...... %3D R in honte N of natural numbers defined (ii) Relation R R = {(x, y): y = x + 5 and x < 4} iii) Relation R in the set A= {1, 2, 3, 4, 5, 6} as R= {(x, y): y is divisible by x} set as holla

Exercise 1.1 Question 1. Determine whether each of the following ręlations are eflexive, symmetric and transitive : (i) Relation R = {(x, y): 3x-y 0} R in the set A= {1, 2, 3, ...13, 14} defined as ...... %3D R in honte N of natural numbers defined (ii) Relation R R = {(x, y): y = x + 5 and x < 4} iii) Relation R in the set A= {1, 2, 3, 4, 5, 6} as R= {(x, y): y is divisible by x} set as holla

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Use first or second differences to determine the type of relation between x and y. 5 -1 2. 1 4. a) linear b) neither linear nor quadratic c) quadratic d) unable to determine 5. The graph of the quadratic relation y = x - 3x – 4 is shown below. For what value or values of x is y = 0? 5. a) x= -1 only b) x= -1 and x = -4 c) x = -4 only d) x= -1 and x = 4 6. The graph of y = x* is translated 3 units to the right. The equation for this new graph is 6. a) y=x² +3 b) y= 3x? c) y =x³ -3 d) y = (x-3)? 7. Decide which direction the graph of y = -x -6 opens, its vertex point, and what its translation from y =x is. 7. a) opens down, vertex at (0, -6), shifted b) opens down, vertex at (-6, 0 ), shifted down 6 units up 6 units c) opens up, vertex at (0, -6), shifted up 6 d) opens up, vertex at (-6,0), shifted units down 6 units 8. Which of the following equations represents an exponential relation? (Ready for Gr.11!? O) 8. a) y=x? b) y=x c) y= 6" d) y =x³ + 2x2 – 5x +1
Consider the relation R = {(1, 1), (1,3), (2,4), (3, 1), (3,3), (4, 3),(4,5)} on the set A = {1,2,3,4,5,6}. Find the Symmetric and reflexive closure.
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11./ a) b) c) 11 1 ) Consider the relation on A={1,2,3) represented by the matrix 0 10 1 Determine if the relation is reflexive. If it is not reflexive, state why. Determine if the relation is symmetric. If it is not symmetric, state why. Sketch a digraph to represent the relation.
Decide whether the following relations is an equivalence relation or not. {(x, y) | xy > 0, x and y are integers}
Exercise 2. Give an example of a relation R on A = {1,2,3} that is neither reflexive or irreflexive.
A.
11./ a) b) c) 11 1 ) Consider the relation on A={1,2,3) represented by the matrix 0 10 1 Determine if the relation is reflexive. If it is not reflexive, state why. Determine if the relation is symmetric. If it is not symmetric, state why. Sketch a digraph to represent the relation.
Consider the relation {(x, y)|x, y E R, (x – 2)² +y² = 1}. (a) Does the above relation represent a function? Graph the relation and justify your answer. (b) Which of the following symmetries are respected by the above relation? symmetry about x-axis, symmetry about y-axis, symmetry about the origin. Justify your answer.