Chapter 3.5, Problem 22E

(a)

To determine

To prove: S be the relation on zz $ℝ\times ℝ$ defined by $\left(x,y\right)S\left(a,b\right)$ if either $x\le a\text{or }y\le b$ . The relation S is antisymmetric.

(b)

To determine

Let A be a set with partial order E and let B and C be subsets of A. If B has an upper bound and C has an upper bound, then $B\cup C$ has an upper bound

(c)

To determine

If B has an upper bound and C has an upper bound, then $B\cup C$ has an upper bound.

(d)

To determine

Let A be a nonempty subset of $ℝ$ , $s=\sup \left(A\right)$ , and $B=\left\{-x:x\in A\right\}$ . Then B is bounded below, and −s = inf(B)

(e)

To determine

Let A be a set with partial order $\le$ . If C $\subseteq$ B $\subseteq$ A and sup (C) exists, then sup (B) exists

(f)

To determine

Let A be a set with a partial order R. If $B\subseteq A$ , u is an upper bound for B, and $u\in B$ , then sup (B) exist, and $u=\sup (B)$

(g)

To determine

Let A be a set with a partial order R. If $B\subseteq A$ , u is an upper bound for B, and $u\in B$ then sup(B) exists and $u=\sup \left(B\right)$

(h)

To determine

Let A be a set with partial order $\le$ If $C\subseteq B\subseteq A$ , $\sup \left(B\right)$ exists, and C is nonempty, then $\sup \left(C\right)$ exists.