please answer all questions in detail For the relation R={(1,3), (1,4), (2,3), (
ID: 3003946 • Letter: P
Question
please answer all questions in detail
For the relation R={(1,3), (1,4), (2,3), (2,4) (3,1), (3,4)} on the set A={1,2,3,4}, explain/shown whether or not the relation is the following: (For any credit, be sure to give a reason why for each). Reflexive, symmetric, antisymmetric, transitive. Let the sets the relations on the all numbers: R_1={a, b) epsilon R^2 a b}, the "greater than or equal to" relation and let R-2={a, b) epsilon R^2 a b}, the "unequal to" relation. Find R_1 cap R_2 (write out the relation in the set notation, as R_1 and R_2 were written) r_1 - R_2 (write out the relation in the set notation, as R_1 and R_2 were written) R_1 R_2 (write out the relation in the set notation, as R_1 and R_2 were written)Explanation / Answer
Answer:
1) Given that A ={1,2,3,4 } and R be the relation defined on the set A such that R ={(1,3),(1,4),(2,3),(2,4),(3,1),(3,4) }
(a) A relation R is reflexive if for every x A then (x,x) R
Hence R is not reflexive since 1 A but (1,1) R
(b) A relation R is symmetric, for any x , y A if (x , y) R then (y , x) R
Hence R is not symmetric since (1,4) R but (4,1) R
(c) A relation R is anti symmetric, for any x , y A if (x , y) R and (y , x) R then a = b.
Hence R is not anti symmetric since (1,3) R and (3,1) R but 1 3.
(d) A relation R is transitive, for any x , y ,z A if (x , y) R and (y , z) R then (x , z) R
Hence R is not transitive since (1,3) R and (3,1) R but (1,1) R.
2) let the sets be relations on the real numbers:
R1 = { (a,b) R2 | a b } and R2 = { (a,b) R2| a b } then
a) R1 R2 = { (a,b) R2 | a > b }
b) R1 - R2 = { (a,b) R2 | a = b }
c) Now R1 - R2 = { (a,b) R2 | a = b } and R2 - R1 = { (a,b) R2 | a < b }
R1 R2 = (R1 - R2) U (R2 - R1) = { (a,b) R2 | a = b } U { (a,b) R2 | a < b }
Therefore, R1 R2 = { (a,b) R2 | a = b or a < b }
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