Describe the equivalence classes generated by T

Question

Suppose $S = \{(x,y) \in \mathbb{R}^2\mid y = x + 1\text{ and } 0 < x < 2\}$.

Question Describe the equivalence relation T on the real line that is the intersection of all equivalence relations on the real line that contain S.

Suppose that $$A_1 = S$$ $$A_2 = \{(x,y) \in \mathbb{R}^2\mid y = x - 1 \text{ and } 1 < x < 3\}$$ $$A_3 = \{(x,y) \in \mathbb{R}^2\mid y = x + 2\text{ and } 0 < x < 1\}$$ $$A_4 = \{(x,y) \in \mathbb{R}^2\mid y = x - 2\text{ and } 2 < x < 3\}$$ $$A_5 = \{(x,y) \in \mathbb{R}^2\mid x - x = 0\}$$

I proved that $T = \bigcup_{i = 0}^{i = 5} A_i$ I am also pretty sure that my proof is correct. My Question is I want to make sure that I am describing the equivalence classes correctly. Let $E_x$ denote the equivalence class for the element $x \in \mathbb{R}$.

If $ -\infty < x \leq 0 \text{ and } 3 \leq x < \infty$, then $E_x = \{x\}$. If $0 < x < 1$, then $E_x = \{x,x + 1,x+2\}$.

If $x = 2,1,$ then $E_x = \{1,2\}$. If $1 < x <2$, then $E_x = \{x,x+1,x - 1\}$. Lastly if $2 < x < 3$, then $E_x = \{x,x - 1,x-2\}$.

Want to make sure I described the equivalence classes correctly.

Answer 1

You're right about $T$ and mostly right about the equivalence classes, with only one definite shortcoming:

• You write "If $ -\infty < x \leq 0 \text{ and } 3 \leq x < \infty$" -- this condition is not satisfied by any $x$. What you mean is, "If $ -\infty < x \leq 0$ or $3 \leq x < \infty$".

Generally, though, I would note that you're not being asked to describe the equivalence class of $x$ for each $x$ -- just to describe all of the equivalence classes, so there's no need to give each of the equivalence classes multiple times just because it has multiple elements. So I would write the result as:

• $\{a\}$ for every $a\in (-\infty,0] \cup [3,\infty)$,
• $\{b,b+1,b+2\}$ for every $b\in(0,1)$, and
• $\{1,2\}$.