1. Let R and S be equlvalence relations on a set A; recall that by definition R, SCAXA. Prove that RnS is an equivalence relation.

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1. Let R and S be equivalence relations on a set A; recall that by definition
R,SCAX A. Prove that RAS is an equivalence relation.
2. Define the relation Ron Z as follows: for r, y EZ, xRy if and only y –x > 0.
Determine whether Ris an equivalence relation, a partial ordering, or neither.
3. Let A = {armadillo, bear, erocodile,dolphin, eagle, fox}. Construct a par-
tial ordering3 on A that satisfies the conditions below. (You may draw a
Hasse diagram to demonstrate your partial ordering.
armadillo 3 eagle.
The least upper bound of the set {bear, eagle, fox} is "crocodile".
"fox" is a minimal element of the partial ordering.
4. Given the slightly blurry Hasse diagram below, find all maximal, minimal,
Transcribed Image Text:1. Let R and S be equivalence relations on a set A; recall that by definition R,SCAX A. Prove that RAS is an equivalence relation. 2. Define the relation Ron Z as follows: for r, y EZ, xRy if and only y –x > 0. Determine whether Ris an equivalence relation, a partial ordering, or neither. 3. Let A = {armadillo, bear, erocodile,dolphin, eagle, fox}. Construct a par- tial ordering3 on A that satisfies the conditions below. (You may draw a Hasse diagram to demonstrate your partial ordering. armadillo 3 eagle. The least upper bound of the set {bear, eagle, fox} is "crocodile". "fox" is a minimal element of the partial ordering. 4. Given the slightly blurry Hasse diagram below, find all maximal, minimal,
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