**Note: We have tried to upload as much as we can, all the question and answers might be shuffled - Please find the answer below each question, some answers might be wrong please review on the last date(some answers might be changed) if you find any wrong answer please comment down below.**

**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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