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Discrete Mathematics
Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B.
(ii) R2 = {(4,y), (y,4)}
(ii) R2 = {(4,y), (y,4)}
Discrete Mathematics
Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B.
(i) R1 = {(2,x), (y,4), (6,z)}
(i) R1 = {(2,x), (y,4), (6,z)}
Discrete Mathematics
Let A be the set {1,2,3,4,5,6}.Which orders pairs are in the following relations on A.
(v) R5 = {(a,b) : a = 2 and a + 2b ≤ 10}
(v) R5 = {(a,b) : a = 2 and a + 2b ≤ 10}
Discrete Mathematics
how to prove the following using defined sets
A=B⇔A⊆B and B ⊆ A
A=B⇔A⊆B and B ⊆ A
Discrete Mathematics
Q2 Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler) (2+1+2+2)
a) Figure out Euler graph from these three graphs.
b) Write down the Euler path of these graphs.
c) If not Euler, provide the reason.
a) Figure out Euler graph from these three graphs.
b) Write down the Euler path of these graphs.
c) If not Euler, provide the reason.
Discrete Mathematics
Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler) (2+1+2+2)
a) Figure out Euler graph from these three graphs.
b) Write down the Euler path of these graphs.
c) If not Euler, provide the reason.
Discrete Mathematics
List the ordered pairs in the equivalence relations produced by these partitions of {a,b,c,d,e,f,g}.
(a) {a,b}, {c,d}, {e,f,g}
(b) {a}, {b}, {c,d}, {e,f}, {g}
(c) {a,b,c,d}, {e,f,g}
(d) {a,c,e,g}, {b,d}, {f}
(a) {a,b}, {c,d}, {e,f,g}
(b) {a}, {b}, {c,d}, {e,f}, {g}
(c) {a,b,c,d}, {e,f,g}
(d) {a,c,e,g}, {b,d}, {f}
Discrete Mathematics
4. (a) Let R be the relation on Z × Z such that ((a,b), (c,d)) ∈ R ↔ a + d = b + c. Show that R is an equivalence relation.
(b) If R is an equivalence relation on a finite non empty set A, then the equivalence classes of R all have the same number of elements?
(c) Prove that intersection of two equivalence relations on a non empty set A is an equivalence relation.
Discrete Mathematics
(b) (i) Let A = {2,4,6} and R = {(2,2), (2,4), (2,6), (4,4), (6,6)} be a relation on A.
Find R−1
.
(ii) Let A = {6,8,10,15} and B = {2,3,4}. Define a relation R from A to B by
(x,y) ∈ R iff x-y is divisible by 2. Find R−1
Find R−1
.
(ii) Let A = {6,8,10,15} and B = {2,3,4}. Define a relation R from A to B by
(x,y) ∈ R iff x-y is divisible by 2. Find R−1
Discrete Mathematics
(a) Let A = {0,1,2}. R = {(0,0), (0,1), (0,2), (1,1), (1,2), (2,2)} and
S = {(0,0), (1,1), (2,2)} be two relations on A.
(i) Show that R is a partial order relation.
(ii) Is R a total order relation?
(iii) Show that S is an equivalence relation.
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S = {(0,0), (1,1), (2,2)} be two relations on A.
(i) Show that R is a partial order relation.
(ii) Is R a total order relation?
(iii) Show that S is an equivalence relation.
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