Relations and Functions Trivia Quiz

Answer: It is transitive

Relations which are reflexive, symmetric and transitive are called "equivalence relations". A relation R on S is reflexive if for every x in S, (x,x) is in R.
A relation R on S is symmetric if for every x,y in S if (x,y) is in R then (y,x) is in R. Transitivity is defined in question 1.

Answer: An equivalence relation.

The definition of an equivalence relation was given in the remarks accompanying question 1. An asymmetric relation is a relation R on set S such that for all x,y in S if (x,y) is in R then (y,x) is NOT in R. To the best of my knowledge I made up the terms "simple relation" and "complex relation". If (x,y) belongs to R, we will write "xRy".

Answer: symmetric but neither transitive nor reflexive.

Since x is never the sibling of x, the relation is not reflexive. If R were symmetric and transitive, xRy (this is shorthand for (x,y) belongs to R) would imply yRx (correct), but then if R were transitive (it is not) xRy and yRx would imply xRx which is not so.

Answer: an equivalence relation.

Each person has the same first name as himself/herself; hence, the relation is reflexive. For any pair of people a and b, if a has the same first name as b, then b has the same first name as a; hence, the relation is symmetric. For any triple of people a, b, and c, if a has the same first name as b and b has the same first name as c, then a has the same first name as c; hence, the relation is transitive.

As indicated in the remarks following question 1, the combined properties of reflexivity, symmetry and transitivity make the relation an equivalence relation.

Answer: This argument is fallacious because there may be an element x in S for which there is no y with xRy.

It is impossible to use symmetry as indicated in the argument if the element x has no partner y for which xRy. Reflexivity requires that for all x in S, xRx.

Answer: This relation is neither reflexive, symmetric nor transitive.

If we had defined R by "(x,y) is an element of R if and only if y is the square of x" then R would be a function. For the relation as defined in the problem, both (16,4) and (16,-4) are elements of R so that the relation R in the problem is not a function.

The domain of a relation or function on a set S is the set of elements a in S for which at least one element b exists in S such that (a,b) is in R. A simple way to describe a function on set S is a relation R on set S with the property that for every a in the domain of S there is exactly one element b in S for which (a,b) is in R.

By the way, the set of elements b in S for which there is at least one element a in S with (a,b) in R is called the range of R.

Answer: is an equivalence relation.

The key is that they speak ALL, and ONLY, the same languages.

Answer: reflexive and symmetric, but not transitive.

To see why transitivity does not hold, imagine three men Jack, Frank and Pierre with the conditions that follow. Jack and Frank both speak English. Frank and Pierre both speak French. Jack does not speak French (or any other language spoken by Pierre).

Answer: neither reflexive, symmetric, nor transitive.

Not everyone loves himself/herself. John loves Mary, but Mary may or may not love John. John loves Mary and Mary loves Tim, but John may or may not love Tim.

Answer: R and R' are each equivalence relations.

Perhaps some day science will find a way for us to have two biological fathers or two biological mothers, but I cannot conceive of it. That's a joke, son.