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Problem 5 Let A, B₁, B2,..., Bn be sets. Prove by induction that for any n ≥ 1, An (UB) = U(An B₂) i=1 i=1 by induction on n.

Problem 5 Let A, B₁, B2,..., Bn be sets. Prove by induction that for any n ≥ 1, An (UB) = U(An B₂) i=1 i=1 by induction on n.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 5 Let A, B₁, B2,..., Bn be sets. Prove by induction that for any n ≥ 1, An (UB) = U(An B₂) i=1 i=1 by induction on n. Hint: Using ellipses, this is equivalent to proving that An (B₁UB₂U...UB₂) = (An B₁) U (An B₂) UU (An B₂). While this is hopefully obvious intuitively (because we've talked about it for n = 2), we are asking you to prove it formally and with an arbitrary number of values with induction. That is, you can use the fact that An (BUC) = (An B) U (ANC), and here we want to prove it more generally. described on the induction Hint: For full marks, use the 7-step process proof paradigm sheet. You'll need to (1) define P(n) that is a boolean function, (1) ensure that P(n) is a function of n, (1) use the correct n as the base case, (1) correctly show that the base case is true, (1) clearly state the induction hypothesis in terms of k (or some variable of your choice), (6) prove the inductive step (1) start with LHS of P(k+ 1) and manipulate to RHS (or vice-versa!). Do not start with LHS = RHS (1) find a way to get the LHS of P(k) in your algebra somewhere, so that you can (1) correctly apply the induction hypotheses, (1) clearly label where the IH is used. (2) find a way to get the RHS of P(k+1) in your algebra somewhere. (1) have a one-sentence conclusion that puts it all together (that we do at the end of every induction proof.)
Transcribed Image Text:Problem 5 Let A, B₁, B2,..., Bn be sets. Prove by induction that for any n ≥ 1, An (UB) = U(An B₂) i=1 i=1 by induction on n. Hint: Using ellipses, this is equivalent to proving that An (B₁UB₂U...UB₂) = (An B₁) U (An B₂) UU (An B₂). While this is hopefully obvious intuitively (because we've talked about it for n = 2), we are asking you to prove it formally and with an arbitrary number of values with induction. That is, you can use the fact that An (BUC) = (An B) U (ANC), and here we want to prove it more generally. described on the induction Hint: For full marks, use the 7-step process proof paradigm sheet. You'll need to (1) define P(n) that is a boolean function, (1) ensure that P(n) is a function of n, (1) use the correct n as the base case, (1) correctly show that the base case is true, (1) clearly state the induction hypothesis in terms of k (or some variable of your choice), (6) prove the inductive step (1) start with LHS of P(k+ 1) and manipulate to RHS (or vice-versa!). Do not start with LHS = RHS (1) find a way to get the LHS of P(k) in your algebra somewhere, so that you can (1) correctly apply the induction hypotheses, (1) clearly label where the IH is used. (2) find a way to get the RHS of P(k+1) in your algebra somewhere. (1) have a one-sentence conclusion that puts it all together (that we do at the end of every induction proof.)
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