Answer to Question #123772 in Discrete Mathematics for Nii Laryea

Question #123772
Let A = {1,2,3,4}. Define a relation R on A by
a R b ↔ a + b ≤ 4
for every a; b ϵ A.
(a) List all the elements of R.
(b) Determine whether R has the following properties. If R has a certain property, prove this
is so, otherwise, provide a counterexample to show that it does not.
i. Reflexivity
ii. Transitivity
iii. Antisymmetry
iv. Symmetry
1
Expert's answer
2020-06-30T18:03:08-0400

Given, A={1,2,3,4}A=\{1,2,3,4\} and R={(a,b):a+b4}.R=\{(a,b):a+b \leq 4\}.


a) The elements of the relation R is, R={(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)}R=\{(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)\}.


b)

i) R is not reflexive, since (3,3)R(3,3)\notin R

ii) R is not transitive, since (2,1)R,(1,3)R(2,1)\in R, (1,3)\in R, but (2,3)R(2,3) \notin R

iii) R is not anti-symmetry, since (1,2)R,(2,1)R(1,2)\in R, (2,1)\in R, but 12.1\neq 2.

iv) For each (a,b)R(a,b)\in R , we have (b,a)R(b,a)\in R. Hence, R is symmetry.


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