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Relations and Functions
Question
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Let f : N – {1} → N defined by f (n) = the highest prime factor of n. Show that f is neither one-to-one nor onto. Find the range of f.

Answer

f : N – {1} → N is defined by
f (n) = the highest prime factors of n.
∴ f (6) = the highest prime factor of 6 = 3 f (12) = the highest prime factor of 12 = 3 Now 6 and 12 are associated to the same element.
∴ f is not one-to-one Also range of f consists of prime numbers only ∴ range of f ≠ N ∴ f is not onto function.
Range of f is the set-of all prime numbers.

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Let A be the set of all students of a boys 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.

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.

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Give an example of a relation which is

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(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.

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 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}

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(v) Relation R in the set A of human beings in a town at a particular time given by
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(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}