1. ## Binary Relations

I need help with binary relations. The question is which properties do the following binary relations satisfy among the reflexive, symmetric, tansitive, irreflexive, and antisymmetric.

a. The congruence relation on the set of triangles.
b. The subset relation on sets.
c. The relation on people that relates people with bachelor's degrees in computer science.
e. The "has a common natinal laguage with" relation on countries.
f. The "is father of " relation on the set of people.

2. Originally Posted by patches
I need help with binary relations. The question is which properties do the following binary relations satisfy among the reflexive, symmetric, tansitive, irreflexive, and antisymmetric.
you know what all these terms mean right? just go through the check list with each part.

example, for the first.

a. The congruence relation on the set of triangles.
Test reflexive:

is it true that every triangle is congruent to itself? YES. Therefore this relation is reflexive

Test symmetric:

is it true that if triangle A is congruent to triangle B, then triangle B is congruent to triangle A (for all triangles A and B)? YES. Thus, this relation is symmetric.

Test transitive:

is it true that if triangle A is congruent to triangle B and triangle B is congruent to triangle C, then triangle A will be congruent to triangle C? YES. So the relation is transitive.

Test Irreflexive:

We already know the relation is reflexive, so it cannot be irreflexive.

You could also note (from previous knowledge) that congruence is an equivalence relation, so we know right off the bat it is reflexive, symmetric and transitive. And so cannot be irreflexive.

Try the others in the same way

3. Hello, patches!

Here's some help . . .

Which properties do the following binary relations satisfy
among: reflexive, symmetric, transitive, irreflexive, and antisymmetric.

a. The congruence relation on the set of triangles.

$\Delta A \cong \Delta A$ . . . True: reflexive

$\text{If }\Delta A \cong \Delta B\text{, then }\Delta B \cong \Delta A$ . . . True: symmetric

$\text{If }\Delta A \cong \Delta B\text{ and }\Delta B \cong \Delta C\text{, then }\Delta A \cong \Delta C$ . . . True: transitive

b. The subset relation on sets.

$A \subseteq A$ . . . True: reflexive

$\text{If }A \subseteq B\text{, then }B \subseteq A$ . . . not true; not symmetric

$\text{If }A \subseteq B\text{ and }B \subseteq A\text{, then }A = B$ . . . True: antisymmetric

$\text{If }A \subseteq B\text{ and }B \subseteq C\text{, then }A \subseteq C$ . . . True: transitive

e. The "has a common national language with" relation on countries.

$A\:\circledR\:A$ . . . True: reflexive

$\text{If }A\:\circledR\:B\text{, then }\,B\:\circledR\:A$ . . . True: symmetric

$\text{If }A\:\circledR\:B\text{ and }B\:\circledR\:C\text{, then }A\:\circledR\:C$ . . . True: transitive

f. The "is father of " relation on the set of people.

$A\:\boxed{F }\:A$ . . . not true: not reflexive

$A\:\boxed{\sim F}\:A$ . . . true: irreflexive

$\text{If }A\:\boxed{F}\:B\text{. then }B\:\boxed{F}\:A$ . . . not true: not symmetric

$\text{If }A\:\boxed{F}\:B\text{ and }B\:\boxed{F}\:C\text{, then }A\:\boxed{F}\:C$ . . . not true: not transitive

4. This is just picking at whoever wrote this question.
Would it be true to say that Canada and England have a common national language?
Would it be true to say that Canada and France have a common national language?