Question

Which of the following are equivalence relations? a. Is similar to for the set T of all triangles in a plane. b. Has the same radius as for the set of

Right triangles and trigonometry

Which of the following are equivalence relations?
a. Is similar to for the set T of all triangles in a plane.
b. Has the same radius as for the set of all circles in a plane.
c. Is the square of for the set N.
d. Has the same number of vertices as for the set of all polygons in a $$\displaystyle{e}.\text{⊆}$$ for the set of sets $$S={A,B,C...}$$
f. "<" for the set R.

(a) A triangle ABC is similar to a triangle DEF if all their angles are equal. This relation is clearly reflexive, symmetric and transitive, so it is an equivalence relation. (b) This relation is reflexive, symmetricand transitive, so it is an equivalence relation. (c) This relation is neither reflexive (2 is not the square of 2), nor symmetric (4is the square of 2 but 2 is not the square of 4) nor transitive (4 is the square of 2 and 16 is the square of 4, but 16 is not the square of 2), so this is not an equivalence relation. (d) This relation is reflexive, symmetricand transitive, so it is an equivalence relation. (e) This relation is reflexive and transitive, but it is not symmetric since {a,b} © {a,b,c} but $${2,8,c}\ ¢\ {a,b}$$, so this is not an equivalence relation. (f) This relation is reflexive and transitive, but it is not symmetric since $$1 \leq 2\ but\ 2$$