Question

40 points question... fastly solve on copy.....

Answer 1

Let R1={(1,1),(2,2),(3,3),(4,4),(2,3),(3,2)(4,2),(4,3)}R1={(1,1),(2,2),(3,3),(4,4),(2,3),(3,2)(4,2),(4,3)}

Step1: R1R1 is reflexive,transitive but not symmetric

Since (a,a)∈Rforalla∈{1,2,3,4}(a,a)∈Rforalla∈{1,2,3,4}

R1R1 is reflexive

Since (4,2)∈Rbut(2,4)∉R(4,2)∈Rbut(2,4)∉R

R1R1 is not symmetric

Since (4,3)∈R,(3,2)∈R=>(4,2)∈R(4,3)∈R,(3,2)∈R=>(4,2)∈R

R1R1 is transitive

R1R1 is reflexive,transitive but not symmetric

Step2: R2R2 is symmetric but neither reflexive nor transitive

Let R2={(1,1),(2,3),(3,2),(1,2)}R2={(1,1),(2,3),(3,2),(1,2)}

Since (2,2),(3,3)(4,4)∉R(2,2),(3,3)(4,4)∉R

R2R2 is not reflexive

Since (2,3)∈R=>(3,2)∈R(2,3)∈R=>(3,2)∈R

R2R2 is symmetric

Since (1,2)∈Rand(2,3)∈Rbut(1,3)∉R(1,2)∈Rand(2,3)∈Rbut(1,3)∉R

R2R2 is not transitive

$R_2 is symmetric but neither reflxive nor transitive

Step3: R3R3 is reflexive,symmetric and transitive

Let R3={(1,1),(2,2),(3,3),(4,4),(2,3),(3,2)(2,4),(4,3),(3,4)(4,2)}R3={(1,1),(2,2),(3,3),(4,4),(2,3),(3,2)(2,4),(4,3),(3,4)(4,2)}

Since (1,1),(2,2)(3,3)(4,4)all∈R3(1,1),(2,2)(3,3)(4,4)all∈R3

R3R3 is reflexive

Since (2,3),(3,2)(2,4)(4,2)(4,3)(3,4)∈R(2,3),(3,2)(2,4)(4,2)(4,3)(3,4)∈R

R3R3 is symmetric

Since (2,3)∈R(3,4)∈R=>(2,4)∈R(2,3)∈R(3,4)∈R=>(2,4)∈R

R3R3 is transitive

R3R3 is reflexive,symmetric and transitive
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Answer 2

$<b><u>$Step1: R1 is reflexive,transitive but not symmetric

Since (a,a)∈Rforalla∈{1,2,3,4}

R1 is reflexive

Since (4,2)∈Rbut(2,4)

R1 is not symmetric

Since (4,3)∈R,(3,2)∈R=>(4,2)∈R(4,3)∈R

R1 is transitive