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Last updated at Sept. 3, 2021 by Teachoo
Transcript
Example 19 Let R be a relation from Q to Q defined by R = {(a, b): a, b ∈ Q and a – b ∈ Z}. Show that (i) (a, a) ∈ R for all a ∈ Q Given R = {(a, b): a, b ∈ Q and a – b ∈ Z} Hence we can say that (a, b) is in relation R if a, b ∈ Q i.e. both a & b are in set Q a – b ∈ Z i.e. difference of a & b is an integer We need to prove both these conditions for (a, a) a, a ∈ Q , i.e. a is in set Q Example 19 Let R be a relation from Q to Q defined by R = {(a, b): a, b ∈ Q and a – b ∈ Z}. Show that (i) (a, a) ∈ R for all a ∈ Q Given R = {(a, b): a, b ∈ Q and a – b ∈ Z} Hence we can say that (a, b) is in relation R if a, b ∈ Q i.e. both a & b are in set Q a – b ∈ Z i.e. difference of a & b is an integer We need to prove both these conditions for (a, a) a, a ∈ Q , i.e. a is in set Q Also, a – a = 0 & 0 is an integer, ∴ a – a ∈ Z Since both conditions are satisfied, (a, a) ∈ R Example 19 Let R be a relation from Q to Q defined by R = {(a,b): a,b ∈ Q and a – b ∈ Z}. Show that (ii) (a, b) ∈ R implies that (b, a) ∈ R Given R = {(a, b): a, b ∈ Q and a – b ∈ Z} Given (a, b) ∈ R , i.e. (a, b) is in relation R . So, the following conditions are true a, b ∈ Q i.e. both a & b are in set Q a – b ∈ Z i.e. difference of a & b is an integer We need to prove both these conditions for (b, a) 1. a, b ∈ Q, hence b, a ∈ Q 2. b – a ∈ Z i.e. difference of b & a is an integer First condition is satisfied as a, b ∈ Q is given For second condition Since negative of integer is also an integer, i.e. –(a – b) is also integer – a + b is integer, b – a is integer b – a is integer ∴ b – a ∈ Z Since both conditions are satisfied, (b, a) ∈ R Example 19 Let R be a relation from Q to Q defined by R = {(a,b): a,b ∈ Q and a – b ∈ Z}. Show that (iii) (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R Given R = {(a, b): a, b ∈ Q and a – b ∈ Z} Given (a, b) ∈ R , i.e. (a, b) is in relation R . So, following conditions are true a, b ∈ Q i.e. both a & b are in set Q a – b ∈ Z i.e. difference of a & b is an integer Given (b, c) ∈ R , i.e. (b, c) is in relation R . So, following conditions are true b, c ∈ Q i.e. both b & c are in set Q b – c ∈ Z i.e. difference of b & c is an integer We need to prove both these conditions for (a, c) 1. Given a, b & b, c ∈ Q, Hence, a, c ∈ Q 2. Given (a – b) & (b – c) are integers, Sum of integers is also an integer So, (a – b) + (b – c) is also integer a – b + b – c is integer, a – c is integer ∴ a – c ∈ Z Since both conditions are satisfied, ∴ (a, c) ∈ R 2. Given (a – b) & (b – c) is an integer, Sum of integers is also an integer So, (a – b) + (b – c) is also integer a – b + b – c is integer, a – c is integer ∴ a – c ∈ Z Since both conditions are satisfied, (a, c) ∈ R
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Example 2
Example 3 Important
Example 4 Important Deleted for CBSE Board 2022 Exams
Example 5 Deleted for CBSE Board 2022 Exams
Example 6 Important
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Example 8 Important
Example 9 Important
Example 10
Example 11 (i)
Example 11 (ii) Important
Example 11 (iii) Important
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Example 14 Important
Example 15 Important
Example 16 Deleted for CBSE Board 2022 Exams
Example 17 Important Deleted for CBSE Board 2022 Exams
Example 18
Example 19 You are here
Example 20 Important
Example 21 Important
Example 22 Important
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