Question

2. (a) (15 points) For each of the following statements about relations on a set A, where $|A| = n$, determine whether the statement is true or false. If it is true, explain briefly why it is true. If it is false, give an example where the statement does not hold. (i) If R is a relation on A and $|R| > n$, then R is reflexive. False $|A| = 2$ and $A = \{x, y\}$ $R = \{(x, y), (y, x)\}$ $|R| = 2$ but R is not reflexive (ii) If $R_1$ and $R_2$ are relations on A with $R_2$ symmetric and $R_1 \subseteq R_2$, then $R_1$ is also symmetric. False Consider the example in 1) $R_2 = R$ (of 1) $R_1 = \{(x, y)\}$ . then $R_2$ is symmetric $R_1$ is not. (iii) If R is an equivalence-relation on A, then $n \le |R| \le n^2$. True, $|R| \ge n$ because it contains all n reflexive pairs. $|R| \le n^2$ because $R \subseteq A \times A$ and $|A \times A| = n^2$

          2. (a) (15 points) For each of the following statements about relations on a set A, where $|A| = n$, determine
whether the statement is true or false. If it is true, explain briefly why it is true. If it is false, give
an example where the statement does not hold.
(i) If R is a relation on A and $|R| > n$, then R is reflexive.
False $|A| = 2$ and $A = \{x, y\}$ $R = \{(x, y), (y, x)\}$
$|R| = 2$ but R is not
reflexive
(ii) If $R_1$ and $R_2$ are relations on A with $R_2$ symmetric and $R_1 \subseteq R_2$, then $R_1$ is also symmetric.
False Consider the example in 1)
$R_2 = R$ (of 1)
$R_1 = \{(x, y)\}$
. then $R_2$ is symmetric
$R_1$ is not.
(iii) If R is an equivalence-relation on A, then $n \le |R| \le n^2$.
True, $|R| \ge n$ because it contains
all n reflexive pairs. $|R| \le n^2$ because $R \subseteq A \times A$
and $|A \times A| = n^2$

$2. (a) (15 points) For each of the following statements about relations on a set A, where |A| = n, determine whether the statement is true or false. If it is true, explain briefly why it is true. If it is false, give an example where the statement does not hold. (i) If R is a relation on A and |R| > n, then R is reflexive. False |A| = 2 and A = {x, y} R = {(x, y), (y, x)} |R| = 2 but R is not reflexive (ii) If R1 and R2 are relations on A with R2 symmetric and R1 ⊆ R2, then R1 is also symmetric. False Consider the example in 1) R2 = R (of 1) R1 = {(x, y)} . then R2 is symmetric R1 is not. (iii) If R is an equivalence-relation on A, then n ≤ |R| ≤ n^2. True, |R| ≥ n because it contains all n reflexive pairs. |R| ≤ n^2 because R ⊆ A × A and |A × A| = n^2$

Added by Renee L.

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