Are these statements correct? If not, state what is wrong. (a) If the relation R

Question

Are these statements correct? If not, state what is wrong.

(a) If the relation R is symmetric and transitive, it is also reflexive. Claim. "Proof" Since R is symmetric, if (x, y)E R, then (y,x) E R. Thus (x, y) E R and (y, x) E R, and since R is transitive, (x, x) E R. Therefore, R is reflexive. Claim. rs is symmetric. "Proof" Suppose (x, y) E R x R. Then (x, y) T (y, x) because x y + x. Therefore, Tis symmetric. Claim. y- s is symmetric. "Proof" Suppose (x,y) and (r, s) are in R × R and (x,y) W(r, s) Then x-r=y-s. Therefore, r_x=s-y, so (r, s) W (x,y). Thus Wis symmetric. (b) The relation T on R × R given by (x,y) T (r, s) iff x+y = (c) The relation W on R × R given by (x,y) W (r, s) iff x-r (d) Claim. If the relations R and S are symmetric, then R n S is symmetric. "Proof." Let R be the relation of congruence modulo 10 and S the relation of congruence modulo 6 on the integers. Both R and S are sym metric. If (x,y) ER n S, then 6 and 10 divide x - y. Therefore, 2, 3, and 5 all divide x - y, so 30 divides x - y. Also if 30 divides x - y, then 6 and 10 divide x - y, so RnS is the relation of congruence modulo 30. Therefore, RnS is symmetric.

Explanation / Answer

Definitions:

R, a relation in a set X, is reflexive if and only if xX, xRx.

R is symmetric if and only if x,yX, xRy yRx.

R is transitive if and only if x,y,zX, xRy yRz xRz.

PART (a) : This statement is wrong. To disprove the claim, here is the counter example :

Take X={0,1,2} and let the relation be {(0,0),(1,1),(0,1),(1,0)}

This is not reflexive because (2,2) isn't in the relation, as for a relation to be reflexive, (0,0) (1,1) (2,2) had to be in the relation for given X.

PART (b) : This statement is true.

(x,y) T (r,s) ---> x+y = r+s

(r,s) T (x,y) ---> r+s = x+y

Addition is commutative. a+b = b+a, and so the statement is true.

PART (c) : This statement is true. It is similar to PART (b).

The proof given below the claim is coherent and correct.

PART (d) : This statement is true.

Suppose R and S are symmetric, and let (a,b) RS. Then because (a,b) R and R is symmetric, (b,a) R. Similarly, (b,a) S, and so (b,a) RS. Since we started with the assumption that (a,b) RS and discovered that (b,a)RS, it follows that RS must be symmetric.

Cheers!

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.