Q4 Show that the intersection of two transitive relations is also transitive. That is, suppose R₁ and R₂ are transitive relations on A, show that R1 R2 is also transitive.

Let A=R0, the set of all nonzero real numbers, and consider the following relations on AA. Decide in each case whether R is an equivalence relation, and justify your answers. (a,b)R(c,d) if and only if ad=bc. (a,b)R(c,d) if and only if ab=cd. (a,b)R(c,d) if and only if a2+b2=c2+d2. (a,b)R(c,d) if and only if ab=cd.

In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.

Does the relation is less than for a numbers have a reflexive property consider one number? A symmetric property consider two numbers? A transitive property consider three numbers?

Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.

Express (AB)(AB) in terms of unions and intersections that involve A,A,B,andB

13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.

21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.

For each of the following relations R defined on the set A of all triangles in a plane, determine whether R is reflexive, symmetric, or transitive. Justify your answers. a. aRb if and only if a is similar to b. b. aRb if and only if a is congruent to b.