Relations and Functions (Part – 1)

Questions Discussed:

Question 1. Let R be the relation in the set N given by
R = {(a, b) : a = b ─ 2, b > 6}. Choose the correct answer.
A. (2, 4) ∈ R
B. (3, 8) ∈ R
C. (6, 8) ∈ R
D. (8, 7) ∈ R (N)

Question 2. Let A= {1, 2, 3} and define R = {(a, b): a ─ b = 12}. Show that R is empty relation on Set A. (B)

Question 3. Let A be the set of all students at a boy's school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R' = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. (N)

Question 4. If A is the set of students at a school then write, which of following relations are Universal, Empty or neither of the two.
R1 = {(a, b) : a, b are ages of students and |a ─ b| > 0}
R2 = {(a, b) : a, b are weights of students, and |a ─ b| < 0}
R3 = {(a, b) : a, b are students studying in same class} (B)

Question 5. Let A = {1, 2, 3} and define R = {(a, b): a + b > 0}. Show that R is a universal relation on set A. (B)

Questions Discussed:

Question 6. Let A = {0, 1, 2, 3} and define a relation R on A as follows:
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}.
Is R reflexive? symmetric? transitive? (E)

Question 7. Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)}.
Choose the correct answer. (N)
A. R is reflexive and symmetric but not transitive.
B. R is reflexive and transitive but not symmetric.
C. R is symmetric and transitive but not reflexive.
D. R is an equivalence relation.

Question 8. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive. (N)

Question 9. Show that the relation R in the set {1, 2, 3} given by
R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive. (N)

Question Discussed:

Question 1. Determine whether each of the following relations are reflexive, symmetric and transitive: (N)
  A. Relation R in the set A\text{ }=\text{ }\left\{ {1,\text{ }2,\text{ }3,\text{ }...,\text{ }13,\text{ }14} \right\} defined as R = \{(x,\text{ }y)\text{ } :\text{ } 3x - y = 0\}
  B. Relation R in the set N of natural numbers defined as R=\left\{ {\left( {x,\text{ }y} \right):y=x+5\text{ and }x < 4} \right\}
  C. Relation R in the set A\text{ }=\text{ }\left\{ {1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6} \right\} as R = \{(x,\text{ } y)\text{ } :\text{ } y\text{ } is\text{ } divisible\text{ } by\text{ } x\}
  D. Relation R in the set Z of all integers defined as R = \{(x,\text{ } y)\text{ } :\text{ } x - y\text{ } is\text{ } an\text{ } integer\}
  E. Relation R in the set A of human beings in a town at a particular time given by
       a. R = {(x, y) : x and y work at the same place}
       b. R = {(x, y) : x and y live in the same locality}
       c. R = {(x, y) : x is exactly 7 cm taller than y}
       d. R = {(x, y) : x is wife of y}
       e. R = {(x, y) : x is father of y}

Questions Discussed:

Question 2. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation. (N)

Question 3. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a ─ b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}. (N)

Question 4. Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive. (N)

Question 5. Show that each of the relation R in the set A=\{x\in Z:0\le x\le 12\} , given by
A. R = {(a, b) : |a ─ b| is a multiple of 4} B. R = {(a, b) : a = b}
is an equivalence relation. Find the set of all elements related to 1 in each case. (N)

Questions Discussed:

Question 6. Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre. (N)

Question 7. Show that the relation R defined in the set A of all triangles as R=\{({{T}_{1}},\,{{T}_{2}}):\,{{T}_{{1\,}}}\,\text{is}\,\text{similar}\,\text{to}\,{{T}_{2}}\} is equivalence relation. Consider three right angle triangles {{T}_{1}} with sides 3, 4, 5, {{T}_{2}} with sides 5, 12, 13 and {{T}_{3}} with sides 6, 8, 10. Which triangles among {{T}_{1}},\,\,{{T}_{2}}\,\,\text{and}\,\,{{T}_{3}} are related? (N)

Question 8. Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5? (N)

Question 9. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2 }. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4. (N)

Question 10. Let A = {0, 1, 2, 3, 4} and define a relation R on A as follows: R = {(0, 0), (0, 4), (1, 1), (1, 3), (2, 2), (3, 1), (3, 3), (4, 0), (4, 4)}. As R is an equivalence relation on A. Find the distinct equivalence classes of R. (P)

Question 11. Let R be the relation on set A=\{x\in Z:0\le x\le 10\} given by R = {(a, b) : (a-b ) is divisible by 4}. Show that R is an equivalence relation. Also, write all elements related to 4. (B)

Question Discussed:

Question 12.Let A = {1, 2, 3, .... , 12} and R be a relation in A\times A defined by (p,\text{ } q)\text{ } \text{R}\text{ } (r,\text{ } s) \text{ if } ps = qr \forall (p,\,\,q),\,\,(r,\,\,s)\,\in \,A\times A . Prove that R is an equivalence relation. Also obtain the equivalence class [(3, 4)]. (B)

Questions Discussed:

Question 13. Let \text{A = \{1, 2, 3, ... 9\}} and R be the relation in A \times A defined by (a,\text{ } b)\text{ R } (c,\text{ } d) \text{ if } a + d = b + c \text{ for } (a,\text{ } b),\text{ } (c,\text{ } d) in A \times A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)]. (B)

Question 14. Let N denote the set of all natural numbers and R be the relation on N \times N defined by (a,\text{ } b)\text{ R } (c,\text{ } d) if ad(b+c)=bc(a+d) . Show that R is an equivalence relation. (B)

Questions Discussed:

Question 15. Show that the relation R in R defined as \text{ R = } \{(a,\text{ } b)\text{ } :\text{ } a ≤ b\}, is reflexive and transitive but not symmetric. (N)

Question 16. Show that the relation R in the set R of real numbers, defined as \text{ R = }\{(a,\text{ } b)\text{ } :\text{ } a ≤ b^2\} is neither reflexive nor symmetric nor transitive. (N)

Question 17. Check whether the relation R in R defined by \text{ R = }\{(a,\text{ } b)\text{ } :\text{ } a ≤ b^3\} is reflexive, symmetric or transitive. (N)