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**Question:**

Let
m be an integer with m > 1. R on the set of integers is an equivalence
relation if

Select
one:

a.
{(a, b) | a ≡ b (mod m)}

b.
{(a, b) | a ≡ a (mod b)}

c.
{(a, b) | b ≡ b (mod a)}

d.
{(a, b) | b ≡ a (mod m)}

**The Correct answer is:**{(a, b) | a ≡ b (mod m)}

**Question:**

Consider
the statement: ∀ x, y E
Z if both xy and x + y are even, then

Select
one:

a.
both x and y are odd

b.
both x and y are even

c.
x is even and y is odd

d.
x is odd and y is even

**The Correct answer is:**both x and y are even

**Question:**

If
A = {0, 1}, B = {1, 2}, and C = {0, 1, 2} then what of the following isn’t in A
× B × C ?

Select
one:

a.
(1, 1, 0)

b.
(2, 2, 0)

c.
(1, 1, 1)

d.
(1, 2, 2)

**The Correct answer is:**(2, 2, 0)

**Question:**

Let
A, B, and C be sets. Identify the the correct one among the following

Select
one:

a.
None of these

b.
A ∩ (B ∩ C) = (C ∪ B) ∪ A

c.
A ∪
(B ∩ C) = (C ∪ B) ∩ B

d.
A ∩ (B ∪
C) = (C ∩ B) ∩ A

**The Correct answer is:**None of these

**Question:**

Let
f1 and f2 be functions from R to R such that f1(x) = x2 and f2(x) = x − x2.
What are

the
functions f1 + f2 and f1 f2?

Select
one:

a.
x2 and x4 – x3

b.
x and x3 – x2

c.
x3 and x2 – x3

d.
x and x3 − x4

**The Correct answer is:**x and x3 − x4

**Question:**

Determine
for what of the following for “f” is not a one-to-one function:

Select
one:

a.
f for f (x) = x + 1 from the set of real
numbers to itself

b.
f from {a, b, c, d} to {1, 2, 3, 4, 5}
with f (a) = 4, f (b) = 5, f (c) = 1, and f (d) = 3

c.
f for f (x) = x2 from Z to the set of
integers

d.
f for f (x) = x2 from Z+ to the set of
integers

**The Correct answer is:**f for f (x) = x2 from Z to the set of integers

**Question:**

Which
of the following is not a logical equivalence for bi-conditional representation
?

Select
one:

a.
￢(p
→ q) ≡ p ∧￢q

b.
p ↔ q ≡ (p ∧ q) ∨ (￢p ∧￢q)

c.
p ↔ q ≡ ￢p
↔￢q)

d.
All mentioned

**The Correct answer is:**￢(p → q) ≡ p ∧￢q

**Question:**

By
the second law of de-Morgan ￢(r ∨ s) is equivalent to

Select
one:

a.
￢r
∧￢s

b.
￢(￢r ∨￢s)

c.
￢r
∨￢s

d.
￢r
∨
s

**The Correct answer is:**￢r ∧￢s

**Question:**

Let
P(x) be the statement “x + 1 > x.” For the real number domain, qualify the
statement for truth value;

Select
one:

a.
∀x∃P(x)
is true

b.
∀xP(x)
is true

c.
∃x∀P(x)
is true

d.
∃xP(x)
is true

**The Correct answer is:**∀xP(x) is true

**Question:**

For
{Z+: Z+ < 5} verify if ∀xP(x) holds good for P(x) is “x2 < 10”

Select
one:

a.
∀xP(x)
is a conjunction

b.
None of these

c.
∀xP(x)
is false

d.
∀xP(x)
holds good

**The Correct answer is:**∀xP(x) is false

**Question:**

Identify
correct statement/s among the following

Select
one:

a.
The relation"Union of sets" is reflexive,but not symmetric

b.
The relation"parallel of lines" is always an equivalence relation

c.
The relation "Division" is Symmetric

**The Correct answer is:**The relation"parallel of lines" is always an equivalence relation

**Question:**

Which
of the following is/are true ?

Select
one or more:

a.
p ∧￢p is always a contradiction

b. ￢p ∨ p is always a tautology

c.
￢p
∨
p is always a contradiction

d.
p ∧￢p is always a tautology

**The Correct answer is:**￢p ∨ p is always a tautology, p ∧￢p is always a contradiction

**Question:**

What
of the following expressions does not imply the negation of the proposition,
“there is an honest politician” if h(x) represents honesty function:

Select
one:

a.
￢∃xH(x)

b.
∃xH(x)

c.
∀x￢H(x)

d.
All of these

**The Correct answer is:**∃xH(x)

**Question:**

“The
sum of two positive integers is always positive” into a logical expression

Select
one:

a.
BOTH :∀x∀y((x
> 0) ∧
(y > 0) → (x +y > 0)) and ∀x∀y(x
+y > 0)

b.
∀x∀y((x
> 0) ∧
(y > 0) → (x +y > 0))

c.
Either ∀x∀y((x
> 0) ∧
(y > 0) → (x +y > 0)) or ∀x∀y(x
+y > 0)

d.
∀x∀y(x
+y > 0)

**The Correct answer is:**BOTH :∀x∀y((x > 0) ∧ (y > 0) → (x +y > 0)) and ∀x∀y(x +y > 0)

**Question:**

Which
of the following is/are statement/s

Select
one or more:

a.
How hot the day is

b.
None of these are statements

c.
The temperature is 40 degrees.

d.
It is raining in the summer.

**The Correct answer is:**The temperature is 40 degrees., It is raining in the summer.

**Question:**

Let
P be “you can take the flight,”

Let
Q be “you buy a ticket.”

What
of the following notates “you can take the flight if and only if you buy a
ticket”

Select
one:

a.
None of these

b.
Q → P

c.
P → Q

d.
P ↔ Q

**The Correct answer is:**P ↔ Q

**Question:**

Consider:
p1 → p2, p2 → p3 and p3 → p1 are true. Then what of the following statements
above integer “n” is not equivalent ?

Select
one:

a.
p3: n2 is even

b.
p2: n − 1 is odd

c.
None of these

d.
p1: n is even

**The Correct answer is:**None of these

**Question:**

How
can we produce the terms of a sequence if the first 10 terms are 5, 11, 17, 23,
29, 35, 41, 47, 53, 59?

Select
one:

a.
6 – 5(n + 1)

b.
5 + 6(n − 1)

c.
5 + 6(n + 1)

d.
6 + 5(n − 1)

**The Correct answer is:**5 + 6(n − 1)

**Question:**

The
compound propositions ￢(p ∨ q) and ￢p ∧￢q are logically equivalent if

Select
one:

a.
￢(p
∨
q) ↔ (￢p ∧￢q) is a tautology

b.
(p ∨
q) ∨
(￢p
∧￢q) results in identity

c.
￢(p
∨
q) ↔ (￢p ∧￢q) is a contradiction

d.
(p ∨
q) ∧
(￢p
∧￢q) is a logical equivalence

**The Correct answer is:**￢(p ∨ q) ↔ (￢p ∧￢q) is a tautology

**Question:**

The
proposition (p ∧ q) → (p ∨
q) is not a tautology for

Select
one:

a.
≡ (￢p
∨
p) ∨
(￢q
∨
q)

b.
≡ (p ∨q)
→ (p ∨ q) ≡ ￢(p ∧ q) ∨ (p ∨ q)

c.
≡ (p ∧
q) → (p ∨ q) ≡ ￢(p ∧ q) ∨ (p ∨ q)

d.
≡ (￢p
∨￢q) ∨ (p ∨ q)

**The Correct answer is:**≡ (p ∨q) → (p ∨ q) ≡ ￢(p ∧ q) ∨ (p ∨ q)

**Question:**

Q(x,
y) being a statement, for what expression of x and y, the truth values of the
propositions Q(1, 2) and Q(3, 0) hold good ?

Select
one:

a.
y = x + 3

b.
x = y + 3

c.
y = x - 3

d.
x = y - 3

**The Correct answer is:**x = y + 3

**Question:**

Consider
the Relation R = {(1,1),(2,3),(3,3),(3,2),(2,2)} on the set A = {1,2,3}.

Which
of the following are true w.r.t to R

Select
one:

a.
R is reflexive but not symmetric and transitive

b.
R is reflexive,symmetric and transitive

c.
R is reflexive and symmetric only but not transitive

d.
R is reflexive and symmetric but not transitive

**The Correct answer is:**R is reflexive,symmetric and transitive

**Question:**

Identify
De-Morgan’s law from the following:

Select
one:

a.
p ∧
(p ∨
q) ≡ p

b.
￢(p
∨
q) ≡ ￢p ∧￢q

c.
￢(p
∧
q) ≡ ￢p ∨ q

d.
p ∨
(p ∧
q) ≡ p

**The Correct answer is:**￢(p ∨ q) ≡ ￢p ∧￢q

thank you so much

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