Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Prove that the following sets are equivalent (a) Intervals (0, 1) and (1,6) (b) What is the cardinality of the following sets? Explain your answer. (i) (1,3) (ii) {x³, x ≤N} (iii) {(m, n) € N× N,m+n≤ 5}
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Transcribed Image Text:Prove that the following sets are equivalent (a) Intervals (0, 1) and (1,6) (b) What is the cardinality of the following sets? Explain your answer. (i) (1,3) (ii) {x³, x ≤N} (iii) {(m, n) € N× N,m+n≤ 5}
* (a) Consider intervals (0.1) and (1,3). Prove that they are equivalent sets. Set F: (01) (03) . 6 761=2x+1 is Bisection? Yes Note: $(x)=3-1x+1 1-0 onto => == + ==> t = w=1 choose t=w_loo Let WE (1,3): It € (0,1) Flt)=w • One-to-one? Yes Let F(x)= F(y) For some 2x+1=2y+1= x=y (b) What is the cardinality of the following sets? Explain your answer. (i) (0,2) U [5,6) (ii) {x²,xN} (iii) {(m,n) Nx N₁ m + n ≤ 4} Uncountable Note: Infinite x,ye (0,1) (i) Cardinality is C Every interval is c and has Cardinality C. (ii) Cardinality is H. Consider F:N →A where A={x²;XENT So, A NN and A Ā=N-K₂ (iii) (m,n)E NXN; m+n ≤4}=A N={ 1, 2, 3, 4, ...} ={(1, 1), (1, 2), (2, 1), (1,3), (3, 1), (2,2) & Note: Finite ANG SO A-G Then F(x)=x² Bijection is Denumerable. Note: Infinite, Countable.
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Transcribed Image Text:* (a) Consider intervals (0.1) and (1,3). Prove that they are equivalent sets. Set F: (01) (03) . 6 761=2x+1 is Bisection? Yes Note: $(x)=3-1x+1 1-0 onto => == + ==> t = w=1 choose t=w_loo Let WE (1,3): It € (0,1) Flt)=w • One-to-one? Yes Let F(x)= F(y) For some 2x+1=2y+1= x=y (b) What is the cardinality of the following sets? Explain your answer. (i) (0,2) U [5,6) (ii) {x²,xN} (iii) {(m,n) Nx N₁ m + n ≤ 4} Uncountable Note: Infinite x,ye (0,1) (i) Cardinality is C Every interval is c and has Cardinality C. (ii) Cardinality is H. Consider F:N →A where A={x²;XENT So, A NN and A Ā=N-K₂ (iii) (m,n)E NXN; m+n ≤4}=A N={ 1, 2, 3, 4, ...} ={(1, 1), (1, 2), (2, 1), (1,3), (3, 1), (2,2) & Note: Finite ANG SO A-G Then F(x)=x² Bijection is Denumerable. Note: Infinite, Countable.
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