



Subscribe to our Youtube Channel - https://you.tube/teachoo
Last updated at Jan. 28, 2020 by Teachoo
Transcript
Example 12 Show that f : N โ N, given by f(x) = {โ(๐ฅ+1 , ๐๐ ๐ฅ ๐๐ ๐๐๐@๐ฅโ1, ๐๐ ๐ฅ ๐๐ ๐๐ฃ๐๐)โค is both one-one and onto. Check one-one There can be 3 cases x1 & x2 both are odd x1 & x2 both are even x1 is odd & x2 is even If x1 & x2 are both odd f(x1) = x1 + 1 f(x2) = x2 + 1 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Putting f(x1) = f(x2) x1 + 1 = x2 + 1 x1 = x2 If x1 & x2 are both are even f(x1) = x1 โ 1 f(x2) = x2 โ 1 If f(x1) = f(x2) x1 โ 1 = x2 โ 1 x1 = x2 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 If x1 is odd and x2 is even f(x1) = x1 + 1 f(x2) = x2 โ 1 If f(x1) = f(x2) x1 + 1 = x2 โ 1 x2 โ x1 = 2 which is impossible as difference between even and odd number can never be even Hence, if f(x1) = f(x2) , Then x1 = x2 โด function f is one-one Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Check onto f(x) = {โ(๐ฅ+1 , ๐๐ ๐ฅ ๐๐ ๐๐๐@๐ฅโ1, ๐๐ ๐ฅ ๐๐ ๐๐ฃ๐๐)โค Let f(x) = y , such that y โ N x = {โ(๐ฆโ1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐@๐ฆ+1, ๐๐ ๐ฆ ๐๐ ๐๐๐)โค If x is odd f(x) = x + 1 y = x + 1 y โ 1 = x x = y โ 1 If x is odd, y is even If x is even f(x) = x โ 1 y = x โ 1 y + 1 = x x = y + 1 If x is even, y is odd Hence, if y is a natural number, x will also be a natural number i.e. x โ N Thus, f is onto.
Examples
Example 2
Example 3
Example 4 Important
Example 5
Example 6 Important
Example 7
Example 8
Example 9
Example 10
Example 11 Important
Example 12 Important You are here
Example 13 Important
Example 14 Important
Example 15 Not in Syllabus - CBSE Exams 2021
Example 16 Not in Syllabus - CBSE Exams 2021
Example 17 Not in Syllabus - CBSE Exams 2021
Example 18 Important Not in Syllabus - CBSE Exams 2021
Example 19 Important Not in Syllabus - CBSE Exams 2021
Example 20 Not in Syllabus - CBSE Exams 2021
Example 21 Not in Syllabus - CBSE Exams 2021
Example 22 Not in Syllabus - CBSE Exams 2021
Example 23 Important Not in Syllabus - CBSE Exams 2021
Example 24 Not in Syllabus - CBSE Exams 2021
Example 25 Important Not in Syllabus - CBSE Exams 2021
Example 26 Not in Syllabus - CBSE Exams 2021
Example 27 Important Not in Syllabus - CBSE Exams 2021
Example 28 Not in Syllabus - CBSE Exams 2021
Example 29 Not in Syllabus - CBSE Exams 2021
Example 30 Not in Syllabus - CBSE Exams 2021
Example 31 Not in Syllabus - CBSE Exams 2021
Example 32 Not in Syllabus - CBSE Exams 2021
Example 33 Not in Syllabus - CBSE Exams 2021
Example 34 Not in Syllabus - CBSE Exams 2021
Example 35 Not in Syllabus - CBSE Exams 2021
Example 36 Not in Syllabus - CBSE Exams 2021
Example 37 Not in Syllabus - CBSE Exams 2021
Example 38 Not in Syllabus - CBSE Exams 2021
Example 39 Not in Syllabus - CBSE Exams 2021
Example 40 Not in Syllabus - CBSE Exams 2021
Example 41 Important
Example 42 Important
Example 43 Important
Example 44
Example 45 Important Not in Syllabus - CBSE Exams 2021
Example 46 Important
Example 47 Important
Example 48 Important
Example 49
Example 50
Example 51 Important
About the Author