i will illustrate what you need to do with this one. the rest of the questions in this section are just applying the definitions again.

**Definition:** A relation

defined on a set

is called

**reflexive** if

for every

(that is, every element in the set is related to itself under the relation).

since

is not always true, each element in the real numbers does not relate to itself under this relation. thus this relation is

**not** reflexive

**Definition: **A relation

defined on a set

is called

**symmetric** if whenever

, then

for all

we are dealing with real numbers, thus we have the property of commutativity of addition.

so if

then

thus we have

as desired. this relation

**is** symmetric

**Definition: **A relation

defined on a set

is called

**anti-symmetric** if whenever

and

then

for all

now if

and

it does not imply that

case in point, consider x = 1 and y = -1

thus, the relation is

**not** anti-symmetric

**Definition:** A relation

defined on a set

is called

**transitive** if whenever

and

then

for all

now if

and

it means that

. but we know that the relation is

**not** reflexive, and therefore

does not relate to itself. so the relation is

**not** transitive.

case in point:

and

then we have

and

(that is, x + y = 1 - 1 = 0, and y + z = -1 + 1 = 0), but

since

. this is an implication, so if we have a true statement implying a false statement, the whole implication is false. thus our conclusion follows

try the rest. as you see, i just relied solely on the definitions, nothing fancy