Prop 7.7) Let f [a, b] R then, 1) L(f, P) ≤U(f, P) 2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)

Consider the set A={a,b,c,d,e,f}. Define a partial ordering on A by setting x<=y if (x,y) belongs to R={(a,a)(b,b)(c,c)(d,d)(e,e)(f,f)(a,d)(a,f)(b,c)(b,d)(e,b)(e,c)(e,d)(f,d)} a)Find all x belongs to A that satisfy x<d b)find all x belongs to A that are incompsctable with c

7. Let A and B be bounded nonempty subsets of R, and let A + B := {a+b : a € A, b € B}. Prove that sup(A + B) = sup A + sup B and inf(A + B) = inf A +inf B.

we defined the symmetric difference of sets A and B to be ΑΔΒ A AB = (A U B) – (A n B) = (A – B) U (B – A). - Prove the associative law for symmetric differences of sets. That is, prove for any sets A, B, and C (ΑΔΒ) ΔC = ΑΔ (BΔ C).