Which of the following relations is reflexive but not transitive for the set T = {7, 8, 9}?

The correct option is R = {(7, 7), (8, 8), (9, 9)}.

To understand why this is the correct choice, let us analyze the properties of a reflexive relation and examine the given options step-by-step.

Reflexive Property:

A relation $R$ on a set $T$ is said to be reflexive if every element of $T$ is related to itself.

In other words, the ordered pair $(x,x)$ must belong to the relation for every element $x$ in the set $T$.

For the set $T=\{7,8,9\}$, any reflexive relation must contain the following pairs:

$(7,7)$, $(8,8)$, and $(9,9)$

Now we evaluate the given options:

• The relation $R=\left\{\right(7,7),(8,8),(9,9\left)\right\}$ contains all three required pairs: $(7,7)$, $(8,8)$, and $(9,9)$. Therefore, it is reflexive.

• The other relations, such as $R=\left\{\right(7,8),(8,7),(8,9\left)\right\}$, $R=\left\{0\right\}$, and $R=\left\{\right(7,8),(8,8),(8,9\left)\right\}$, do not contain all the required reflexive pairs and are therefore not reflexive.

Since the relation R = {(7, 7), (8, 8), (9, 9)} is the only option that satisfies the reflexive property, it is the correct choice.