Question

Construct a 3 x 4 matrix, whose elements are given by :

$straight a subscript straight i space straight j end subscript equals 1 half space vertical line space minus 3 space straight i space plus space straight j right square bracket$

Solution

Let A = [a_{i j} be required 3x4 matrix where

a subscript i j equals 1 half vertical line minus 3 space i space plus space j space i vertical line end subscript

therefore space space space space straight a subscript 11 equals 1 half space space left square bracket space minus 3 plus 1 right square bracket space equals 1 half left parenthesis 2 right parenthesis equals 1 comma space space straight a subscript 12 equals 1 half vertical line minus 3 plus 2 vertical line equals 1 half left parenthesis 1 right parenthesis equals 1 half space space space space space space straight a subscript 13 equals 1 half vertical line minus 3 plus 3 vertical line equals 1 half left parenthesis 0 right parenthesis equals 0. space straight a subscript 14 equals 1 half vertical line minus 3 space plus space 4 vertical line equals 1 half left parenthesis 1 right parenthesis equals 1 half space space space space space space straight a subscript 21 equals 1 half space vertical line minus 6 plus 2 vertical line equals 1 half left parenthesis 5 right parenthesis equals 5 over 2. straight a subscript 22 equals 1 half space vertical line minus 6 plus 2 right curly bracket equals 1 half left parenthesis 4 right parenthesis equals 2 space space space space space space straight a subscript 23 equals 1 half space vertical line minus 6 minus 3 vertical line equals 1 half left parenthesis 3 right parenthesis equals 3 over 2. space straight a subscript 24 equals 1 half space vertical line minus 6 plus 4 vertical line equals 1 half left parenthesis 7 right parenthesis equals 7 over 2 space space space space space space straight a subscript 33 equals 1 half space vertical line minus 9 plus 3 vertical line space equals space 1 half space left parenthesis 6 right parenthesis space equals 3. space straight a subscript 34 equals 1 half space vertical line minus 9 plus 4 vertical line equals 1 half left parenthesis 5 right parenthesis equals 5 over 2 therefore space space straight A equals open square brackets table row 1 cell 1 half end cell 0 cell 1 half end cell row cell 5 over 2 end cell 2 cell 3 over 2 end cell 1 row 4 cell 7 over 2 end cell 3 cell 5 over 2 end cell end table close square brackets

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}

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Relations And Functions

Construct a 3 x 4 matrix, whose elements are given by :

$straight a subscript straight i space straight j end subscript equals 1 half space vertical line space minus 3 space straight i space plus space straight j right square bracket$

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.

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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For Daily Free Study Material Join wiredfaculty

Relations And Functions

Construct a 3 x 4 matrix, whose elements are given by :

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.

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.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Construct a 3 x 4 matrix, whose elements are given by :

$straight a subscript straight i space straight j end subscript equals 1 half space vertical line space minus 3 space straight i space plus space straight j right square bracket$

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.

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