Discrete Mathematics Answers

Let A = {1, 2, 3, 4}. Define a relation R on A by

a R b ⇐⇒ a + b ≤ 4

for every a, b ∈ 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

Determine whether the statement p→(q∨r) is equivalent to (p∧∼r)→q ?

Let X= {1, 2, 3, 4}Y= {a, b, c} and Z={1,2}

1. Define K: Y→X as follows: K(a) = 1, K(b) = 2, K(c) = 3, and K(c) = 4. Is K onto? If not explain.

2. Using arrow diagram, find one function from X to Y that is Onto but not One-to-One

3. Define a function G: Y→Z that is onto and one-to-one and represent through matrix

If p→q is false can you find the truth value of ~ (p ∧ q) →q? Explain your answer.

Let X= {1, 2, 3, 4}Y= {a, b, c} and Z={1,2}

1. Define K: Y→X as follows: K(a) = 1, K(b) = 2, K(c) = 3, and K(c) = 4. Is K onto? If not explain.

2. Using arrow diagram, find one function from X to Y that is Onto but not One-to-One

3. Define a function G: Y→Z that is onto and one-to-one and represent through matrix.

Part (a): Let A and B are any sets then show that A-(A∩B)=(A∩A^c)∪(A∩B^c) by using membership table.

Part (b): Draw Venn diagram to describe sets A, B, and C that satisfy the given conditions.


A∩B≠ϕ,B∩C≠ϕ,A∩C=ϕ,A⊈B,C⊈B.

: Let A = {1, 2, 3, 4, 5} then define a relation R on A as (a.b)∈R iff a≤b and Relation Ton A as (a,b)∈T iff a/b.


Represent R by an matrix.

Is R Reflexive? Transitive? Give a valid reason for your answer.

Is T Antisymmetric? Give a valid reason for your answer.

Represent T by an matrix

Represent T by an arrow diagram.

Let X= {1, 2, 3, 4}Y= {a, b, c} and Z={1,2}

1. Define K: Y→X as follows: K(a) = 1, K(b) = 2, K(c) = 3, and K(c) = 4. Is K onto? If not explain.

2. Using arrow diagram, find one function from X to Y that is Onto but not One-to-One

3. Define a function G: Y→Z that is onto and one-to-one and represent through matrix.

Let A= {1, 2, 3, 4, 5}then define a relation R on A as (a.b)∈R iff a≤b and Relation Ton A as (a,b)∈T iff a/b.


Represent R by a matrix.

Is R Reflexive? Transitive? Give a valid reason for your answer.

Is T Antisymmetric? Give a valid reason for your answer.

Represent T by an matrix

Represent T by an arrow diagram.

Part (a): Let Aand B are any sets then show that A-(A∩B)=(A∩A^c)∪(A∩B^c) by using membership table.

Part (b): Draw Venn diagram to describe sets A, B, and C that satisfy the given conditions.


A∩B≠ϕ,B∩C≠ϕ,A∩C=ϕ,A⊈B,C⊈B.

Part (c): Find negation of the following statement: If cows are crows then crows are four legged.