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(i) Between any two distinct rational numbers, there are infinitely many irrationals. (ii) There is a bijection from the middle-thirds Cantor set C to R+. (iii) If A is uncountable, than A~ R. Recall that A B means that there is a bijection f A → B. (iv) Every non-empty SCR that is bounded from below has a greatest lower bound. (v) For some fixed universal set U, the relation is a partial order on P(U). Recall that A B means that there is an injection f : A → B. (vi) There is a bijection R → R2, where R2 = R x R is the usual Euclidean plane.

(i) Between any two distinct rational numbers, there are infinitely many irrationals. (ii) There is a bijection from the middle-thirds Cantor set C to R+. (iii) If A is uncountable, than A~ R. Recall that A B means that there is a bijection f A → B. (iv) Every non-empty SCR that is bounded from below has a greatest lower bound. (v) For some fixed universal set U, the relation is a partial order on P(U). Recall that A B means that there is an injection f : A → B. (vi) There is a bijection R → R2, where R2 = R x R is the usual Euclidean plane.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.4: Binary Operations
Problem 12E
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True or false. (no justification needed)

(i) Between any two distinct rational numbers, there are infinitely many irrationals. (ii) There is a bijection from the middle-thirds Cantor set C to R+. (iii) If A is uncountable, than A~ R. Recall that A B means that there is a bijection f A → B. (iv) Every non-empty SCR that is bounded from below has a greatest lower bound. (v) For some fixed universal set U, the relation is a partial order on P(U). Recall that A B means that there is an injection f : A → B. (vi) There is a bijection R → R2, where R2 = R x R is the usual Euclidean plane.
Transcribed Image Text:(i) Between any two distinct rational numbers, there are infinitely many irrationals. (ii) There is a bijection from the middle-thirds Cantor set C to R+. (iii) If A is uncountable, than A~ R. Recall that A B means that there is a bijection f A → B. (iv) Every non-empty SCR that is bounded from below has a greatest lower bound. (v) For some fixed universal set U, the relation is a partial order on P(U). Recall that A B means that there is an injection f : A → B. (vi) There is a bijection R → R2, where R2 = R x R is the usual Euclidean plane.
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