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Relations And Functions

Question
CBSEENMA12032428
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Find the values of.x, y and z from the following equations :  open square brackets table row cell straight x plus straight y space space space space end cell 2 row cell 5 plus straight z space space space space end cell cell straight x space straight y end cell end table close square brackets equals space open square brackets table row 6 2 row 5 8 end table close square brackets

Solution
We are given that  open square brackets table row cell straight x plus straight y space space space space end cell 2 row cell 5 plus straight z space space space space end cell cell straight x space straight y end cell end table close square brackets equals space open square brackets table row 6 2 row 5 8 end table close square brackets

By definition of equality of matrices,
x+.y= 6    ....(1)
5 + z =5    ....(2)
x y =8    ...(3)
From (2). z= 0
From (I). y= 6 – x   ...(4)
Putting y = 6 –x in (3). we get.
x(6 – x) = 8 or 6 x – x2 –8 = 0
∴x2–6x + 8 = 0 ⇒ (.x –2) (x–4) = 0;⇒ .x=2.4
;∴ from (4). y = 6 – 2. 6 – 4 = 4, 2
∴we have
x = 2, ,y = 4, z = 0 ; .x = 4, y = 2, z = 0

Some More Questions From Relations and Functions Chapter

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

 Determine whether each of the following relations are reflexive, symmetric and transitive :

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as 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} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(v) 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}

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.

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

Check whether the relation R in R defined by R = {(a,b) : a ≤ b3} is refleive, symmetric or transitive.

If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’. 

If R and R’ are symmetric relations on a set A, then R ∩ R’ is also a sysmetric relation on A.

Show that the union of two symmetric relations on a set is again a symmetric relation on that set.

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