Discrete Mathematics Answers

show that

are logically equivalent

Draw a graph which has an Euler circuit but is not planar. Formalize the graph in the form 

Draw a graph which does not have an Euler path and is also not planar. Formalize the graph in the form 

Note: If you cannot draw the graph due to technical reasons, it is OK to just use formal notation and describe the graph textually.

.4 If Universal Set U = {90, 91 , 92 , 93 , 94, 95 , 96 , 97 , 98, 99 , 100} (10)

A = {90, 92, 94, 96, 98, 100}, B= {91, 93, 95, 97, 99},

C = {90, 94, 98}

1.4.1 What is (A ∩ C)c 1.4.2 What is (B ∪ C

give steps please

1.3 Using a Truth table, determine the value of the compound proposition

((𝑝 ∨ 𝑞) ∧ (¬𝑝 ∨ 𝑟)) → (𝑞 ∨ 𝑟)

give steps please

1.1 Determine whether ( 𝑝∨𝑞)∧(𝑝→𝑟)∧( 𝑞→𝑠)→𝑟∨𝑠 is a Tautology or a contradiction

And give steps please

Let R1 and R2 be two relation on real number such that R1 = {(x, y)|x < y} and R2 =

{(x, y)|x > y}, then find R1 ∪ R2,R1 ∩ R2,R1 − R2, R2 − R1, and R1+LR2.

Suppose a, b, c, d have proper positions 1, 2, 3, 4 respectively, i.e., the cor-

rect sequence (from position 1 to 4) is a, b, c, d. Write down all the deranged

sequences. What is the combinatorial expression for their count?

For each of the ff. sets, determine whether 2 is an element of that set.

1. {{2},{2,{2}}}
2. {{{2}}}

Express each of these mathematical statements using predicates, quantifiers, logical connectives, and mathematical operators.

a) The product of two negative real numbers is positive.

b) The difference of a real number and itself is zero.

c) Every positive real number has exactly two square roots.

d) A negative real number does not have a square root that is a real number. e) Every non-zero real number has a unique reciprocal. 

Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers.

a) ∃x ∀y(x + y = y)

b) ∀x ∀y (((x ≥ 0) ∧ (y < 0)) → (x − y > 0))

c) ∃x ∃y(((x ≤ 0) ∧ (y ≤ 0)) ∧ (x − y > 0))

d) ∀x ∀y((x ≠ 0) ∧ (y ≠ 0) ↔ (xy ≠ 0))