Consider the set X to be a collection of colored shaped objects, where X = {red
ID: 3037175 • Letter: C
Question
Consider the set X to be a collection of colored shaped objects, where X = {red triangle, blue square, yellow circle, red circle, yellow triangle, green circle}. For a, b elementof X, let's say a ~ b if object a is the same shape as object b. (a) Prove that ~ is an equivalence relation. (For example, "a ~ a because a has the same shape as itself") (b) List the equivalence classes of ~ on X. Describe them, i.e. for each class, say what's in common among all of its elements. (c) Why is[triangle]not an equivalence class on this set X?Explanation / Answer
For notational convenience , let R be the given relation.
By definition, aRb if a has the same shape as b.
(a) Claim : R is an equivalence relation
(1) R is reflexive: aRa for a, as a and a have the same shape
(2) R is symmetric: aRb implies a has the same shape as b , which means b has the same shape as a
which in trun means bRa.
(3) R is transitive: If aRb and bRa, then clearly a and have the same shape, so aRc is true.
Hence R is an equivalence relation.
(b) List all equialence classes.
C = { a: a is a circle}
T = {a: a is a triangle}
S = {a: a square}
(c) [triangle ] is indeed an equivalence class , as noted above.