Check all properties that the following relations on the set of positive integers have. The relations are defined by (x, y)∈R if and only if the indicated condition is satisfied.

(a) x and y have a common prime factor

A. symmetric
B. reflexive
C. transitive
D. antisymmetric
E. irreflexive

(b) x−y is a multiple of 3

A. irreflexive
B. reflexive
C. transitive
D. symmetric
E. antisymmetric

(c) x and y are both prime

A. symmetric
B. irreflexive
C. transitive
D. antisymmetric
E. reflexive

(d) x and y are twin primes (they are primes and their distance is 2)

A. irreflexive
B. antisymmetric
C. reflexive
D. transitive
E. symmetric

< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)

step-by-step explanation:

ιт ιѕ $222.95

step-by-step explanation:

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тнєи ѕυвтяα¢т ιт fяσм $318.50-$95.55=$222.95

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