Solution (a) R = {(a,b) = (Z>0)²: Ak € Z such that a o. There can exist no real number c such that a o satisfy (a,b) € R. Then there is no k € Z such that a

Dedekind Cuts Notes from Principles by Rudin Definition A set ack is a cut if LA I) a contains at least one rational number but not every rational, (II) If pea, q

Find he alative morima and minima, H any tar the fuechon If there is not an answer wite /A Wade ms wefo as albered paics.

8. Suppose the variables x and y represent real numbers, and L(x, y) : x < y, Q(x, y) : x = y, E(x) : x is even, I(x) : x is an integer. Write the statement using these predicates and any needed quantifiers. a) Every integer is even. b) If x < y, then x is not equal to y. c) There is no largest real number.

Use the following definitions and the laws of logic to prove the following statement. • Definition of Cartesian product: (x, y) E Ax B + (x€ A) ^ (y E B) • Definition of union: x E AUB → (x E A) V (x € B) Statement. For any sets A, B, C, A × (BUC) = (A × B) U (A × C).

What is the size of the symmetric difference of two finite sets A and B, that is AA BI ? o IIAI - |B|| O IAl +|B| – 2|A B| O IAl +|B| – |AN BỊ O IA| + |B|

Show that CC R" is closed if and only if bdry (C) C C.