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Laim. {21n: ne Z} U (14n: ne Z} C {7n: ne Z}. a) ( -) Write the claim as an (equivalent) if-then statement. (., Give a direct proof by cases that the claim is true. As a hint, you might want to prove the if-then statement you constructed in (a). D( .) State (but do not prove) the contrapositive of your statement from part (a). (..) State (but do not prove) the converse of your statement from part (a). (, Give a disproof by counter-example of the converse from part (d). (That is, show that the converse is not true by providing an example that demonstrates it is not true.)

Laim. {21n: ne Z} U (14n: ne Z} C {7n: ne Z}. a) ( -) Write the claim as an (equivalent) if-then statement. (., Give a direct proof by cases that the claim is true. As a hint, you might want to prove the if-then statement you constructed in (a). D( .) State (but do not prove) the contrapositive of your statement from part (a). (..) State (but do not prove) the converse of your statement from part (a). (, Give a disproof by counter-example of the converse from part (d). (That is, show that the converse is not true by providing an example that demonstrates it is not true.)

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 1 Consider the following claim: Claim. {21n: n € Z} U {14n: ne Z} c{7n: ne Z}. (a) () Write the claim as an (equivalent) if-then statement. (b) ( . Give a direct proof by cases that the claim is true. As a hint, you might want to prove the if-then statement you constructed in (a). (c) (' .) State (but do not prove) the contrapositive of your statement from part (a). (d) (..) State (but do not prove) the converse of your statement from part (a). (e) (... Give a disproof by counter-example of the converse from part (d). (That is, show that the converse is not true by providing an example that demonstrates it is not true.)
Transcribed Image Text:Problem 1 Consider the following claim: Claim. {21n: n € Z} U {14n: ne Z} c{7n: ne Z}. (a) () Write the claim as an (equivalent) if-then statement. (b) ( . Give a direct proof by cases that the claim is true. As a hint, you might want to prove the if-then statement you constructed in (a). (c) (' .) State (but do not prove) the contrapositive of your statement from part (a). (d) (..) State (but do not prove) the converse of your statement from part (a). (e) (... Give a disproof by counter-example of the converse from part (d). (That is, show that the converse is not true by providing an example that demonstrates it is not true.)
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