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5. Prove that if S and I are countably infinite sets (meaning |S| = |N| = |T|) then SUT is also countably infinite. Hint: Start by proving this in the case where S and T are disjoint, SnT=0, and then prove it in the general case. It might be helpful, although not necessary, to use the Cantor-Schröder- Bernstein theorem, mentioned in Lecture 2.

5. Prove that if S and I are countably infinite sets (meaning |S| = |N| = |T|) then SUT is also countably infinite. Hint: Start by proving this in the case where S and T are disjoint, SnT=0, and then prove it in the general case. It might be helpful, although not necessary, to use the Cantor-Schröder- Bernstein theorem, mentioned in Lecture 2.

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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5. Prove that if S and I are countably infinite sets (meaning |S| = |N| = |T|) then SUT is also countably infinite. Hint: Start by proving this in the case where S and T are disjoint, SnT = 0, and then prove it in the general case. It might be helpful, although not necessary, to use the Cantor-Schröder- Bernstein theorem, mentioned in Lecture 2. For the remaining exercises, (F, +,,<) is an ordered field. You may use any of the axioms and eorems stated in class about ordered fields. 6. Prove that Va € F, -(-a)= a. 7. Prove that Va, b = F, (-a) b= -(a - b). 8. Prove that Va € F, (a a) +1> 0. 1 9. Use the previous result to prove that it is impossible to put an ordering on the field of complex numbers (C, +,-) that would satisfy the ordered field axioms, (01)-(04) from lecture.
Transcribed Image Text:5. Prove that if S and I are countably infinite sets (meaning |S| = |N| = |T|) then SUT is also countably infinite. Hint: Start by proving this in the case where S and T are disjoint, SnT = 0, and then prove it in the general case. It might be helpful, although not necessary, to use the Cantor-Schröder- Bernstein theorem, mentioned in Lecture 2. For the remaining exercises, (F, +,,<) is an ordered field. You may use any of the axioms and eorems stated in class about ordered fields. 6. Prove that Va € F, -(-a)= a. 7. Prove that Va, b = F, (-a) b= -(a - b). 8. Prove that Va € F, (a a) +1> 0. 1 9. Use the previous result to prove that it is impossible to put an ordering on the field of complex numbers (C, +,-) that would satisfy the ordered field axioms, (01)-(04) from lecture.
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