Confused on how to do problems 1 and 7 7b Which of the prope

Confused on how to do problems 1 and 7?

7b.) Which of the properties reflexive, symmetric, transitive R possess?

CHAPTER 5 HIGHLIGHTS NS Supplementary Exercises for Chapter 5 193 n integer n 2 2, let S 11,2, n and let T, be the set of all 2-element subsets of l. For For A, B e Tn, define the relation Rn on Tn by A Rn B if An B =0. For n = 4, list all 1. For an i ra elements of Rn = (a, b). How many relations on A are there? 2. Let A fa, b). How mar Let A = { i, 2.3.4). For the relation R (1,1), (2,2),(2,3), (3,2), (4.4)) on A, determine which of the properties reflexive, symmetric, transitive R possesses. Let A = {1,2,3) and let R-{(1,3)) be a relation on A. Which of the properties reflexive. symmetric, transitive does R possess? 4. Let A = 5. Let R be the relation defined on Z by a R b if a bor a 2b (a) Give an example of two integers that are related by R and two integers that are not. (b) Which of theproperties reflexive, symmetric, transitive does R possess? 6. A relation R is defined on N by a R b if b-a\" for some n E N. (a) Give an example of two positive integers that are related by R and two positive integers that are not. (b) Which of the properties reflexive, symmetric, transitive does R possess? 7. A relation R is defined on R by a R b if ab 0. (a) Give an example of two real mumbers that are related by R and two real numbers that are not. itive does R possess?

Solution

Hey, Lets take your questions one by one: ( I will answer upto 4, as per time), 5-7 can be done based on my below explanations:

1.) For n=4:

S4={1,2,3,4}, then Tn (all elements subsets of S4)={(1,2),(1,3),(1,4),(2,1),(2,3),(2.4),(3,1)(3,2)(3,4)(4,1)(4,2)(4,3),(1,1),(2,2),(3,3)(4,4)}

Now R on T (no Common elements in A,B) => {(1,1),(2,2),(3,3)(4,4)}=> A reflexive relation. max elements.

Consider an instant : A=(1,1), B=(2,2) A intersection B has no elements.

2.)  The relations on a set A are the subsets of A x A. Here A has 2 elements then A x A has 2^2 = 4 elements. The number of subsets i.e, relations on A, is 2^4 = 16. Formally, the number of relations on a set with n elements is 2^{n^2}. = 2*(2^2)=16.

3) Reflexive is (A A), Symmetric [(A B) (B A)], Transitive [ (A B) (B C)=> (A C)]

Now, R is reflexive if for all x belongs A, (x,x) belongs R. In this (3 3) (4 4) doesn\'t belong to R so NOT Reflexive

Transitive yes 2 2 , 2 3=> 2 3 belongs R, 2 3 , 3 2 => 2 2 belongs R so YES, symmetric.

Symmetric yes beause if 2 3 in R then 3 2 is also there in R

4) Baesd on Explanation given in 3rd statement above, for (1,2,3)=> R={(1,3)} is NOT REFLEXIVE , Neither symmetric ( 3,1 ) NOT there,

BUT YES it is transitive , there are no counter example which can prove it is NOT.