Determine whether the given relation is reflexive, symmetric and transitive: C i

Question

Determine whether the given relation is reflexive, symmetric and transitive: C is the circle relation on the set of real numbers: For all x y Element R x C y x^2 + y^2 = 1. D is the "divides" relation on Z^+: For all positive integers m and n m D n m|n. Define a relation Q on R as follows: For all real numbers x and y x Q y x - y is rational. Let A he the set of all strings of a s and b's of length 4. Define a relation R on A as follows: For all s. t Element A. s R t s has the same first two characters as t. Let A he the set of all strings of 0's. l's and 2's of length 4. Define a relation R on A as follows: For all s. t Element A. s R t the sum of the characters in v equals the sum of the characters in t. Let A be the set of all English statements. A relation I is defined on A as follows: For all p q Element A. p | q p rightarrow q is true.

Explanation / Answer

6)

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Reflexive defination:-In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.

let l contain ={(p,q),(q,r),(p,p),(p,r)...}

for reflexive p l p <-> p->p it is always true so

for all x belongs to A we have a relation (x,x) or x l x in ' l ' relation so IT IS REFLEXIVE

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Symmtric property :-a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a

so let l relation contain (p,q)

that means p -> q is true

so for symmtricity it should be contain (q,p) for this q->p must be true . but it is not always true some time true and some tirme falls so it is not Symmtric relation

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Transitive:-

let l contain (p,q)(q,r)

that means p->q true

q-> r true

so by defination of transitive property 'l' should be contain (p,r) relation but for this p->r must be true.

so we know that by property of ->(implication)

if p->q ,q->r then p->r always hold that means always true so

(p,r) must be come in relation in 'l' so it is Transitive

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