For each part, give a relation that satisfies the condition.

1. Reflexive and symmetric but not transitive
2. Reflexive and transitive but not symmetric
3. Symmetric and transitive but not reflexive

• A relation R is said to be reflexive if$(\mathrm{a},\mathrm{a})\in \mathrm{R}$ for every$\mathrm{a}\in \mathrm{A}$
• A relation R is said to be symmetricif $(\mathrm{a},\mathrm{b})\in \mathrm{R}$then$(\mathrm{b},\mathrm{a})\in \mathrm{R}$ for every$\mathrm{a},\mathrm{b}\in \mathrm{A}$
• A relation R is said to be transitive if $(\mathrm{a},\mathrm{b})\in \mathrm{R}$and$(\mathrm{b},\mathrm{c})\in \mathrm{R}$ then$(\mathrm{a},\mathrm{c})$ must belong to R for every$\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{A}$.

$\mathrm{R}=\{(4,4),(6,6),(8,8),(4,6),(6,4),(6,8),(8,6)\}$

The relation R is reflexive because the set contains three elements 4, 6, and 8, so the relation must have$(4,4),(6,6),(8,8)$ ordered pairs in the relation R to be reflexive, which is true. So, it is reflexive.

The relation R is symmetric because for every$(\mathrm{a},\mathrm{b})$ , there is correspondingrole="math" localid="1658132970308" $(\mathrm{b},\mathrm{a})$ that belongs to R.

The relation R is not transitive because it does not satisfy the property of transitivity. For instance, we have$\left(4,6\right)$ and$\left(6,8\right)$ in the relation R. So, to satisfy the transitivity $(4,8)\in \mathrm{R}$, should satisfy, which is not.

$\mathrm{R}=\{(4,4),(6,6),(8,8),(4,6)\}$

The relation R is reflexive because the set contains three elements 4, 6, and 8 so the relation must have $(4,4),(6,6),(8,8)$ordered pairs in the relation R to be reflexive, which is true. So, it is reflexive.

The relation R is not symmetric because for every $(\mathrm{a},\mathrm{b})$, there should be corresponding $(\mathrm{b},\mathrm{a})$that belongs to R, which is not true because $(4,6)\in R$but $(6,4)\notin \mathrm{R}$.

The relation R is transitive because for all ordered pairs$(\mathrm{a},\mathrm{b})$ and $(\mathrm{b},\mathrm{c})$, there is an ordered pair$(\mathrm{a},\mathrm{c}).$

$\mathrm{R}=\left\{(4,4),(6,6)\right\}$

The relation R is not reflexive because the set contains three elements 4, 6, and 8 so the relation must have $(4,4),(6,6),(8,8)$ordered pairs in the relation R to be reflexive, which is not true. So, it is not reflexive.

The relation R is symmetric because for every$(\mathrm{a},\mathrm{b})$ , there is corresponding$(\mathrm{b},\mathrm{a})$ that belongs to R.

The relation R is transitive because for all ordered pairs$(\mathrm{a},\mathrm{b})$ and $(\mathrm{b},\mathrm{c})$, there is an ordered pair$(\mathrm{a},\mathrm{c}).$