1
GATE CSE 2023
MCQ (More than One Correct Answer)
+2
-0.67

Let $$f:A \to B$$ be an onto (or surjective) function, where A and B are nonempty sets. Define an equivalence relation $$\sim$$ on the set A as

$${a_1} \sim {a_2}$$ if $$f({a_1}) = f({a_2})$$,

where $${a_1},{a_2} \in A$$. Let $$\varepsilon = \{ [x]:x \in A\}$$ be the set of all the equivalence classes under $$\sim$$. Define a new mapping $$F:\varepsilon \to B$$ as

$$F([x]) = f(x)$$, for all the equivalence classes $$[x]$$ in $$\varepsilon$$.

Which of the following statements is/are TRUE?

A
F is NOT well-defined.
B
F is an onto (or surjective) function.
C
F is a one-to-one (or injective) function.
D
F is a bijective function.
2
GATE CSE 2023
MCQ (More than One Correct Answer)
+2
-0.67

Let X be a set and 2$$^X$$ denote the powerset of X. Define a binary operation $$\Delta$$ on 2$$^X$$ as follows:

$$A\Delta B=(A-B)\cup(B-A)$$.

Let $$H=(2^X,\Delta)$$. Which of the following statements about H is/are correct?

A
H is a group.
B
Every element in H has an inverse, but H is NOT a group.
C
For every $$A\in2^X$$, the inverse of A is the complement of A.
D
For every $$A\in2^X$$, the inverse of A is A.
3
GATE CSE 2021 Set 1
MCQ (More than One Correct Answer)
+2
-0.67
A relation R is said to be circular if aRb and bRc together imply cRa. Which of the following options is/are correct?
A
If a relation S is transitive and circular, then S is an equivalence relation.
B
If a relation S is reflexive and symmetric, then S is an equivalence relation.
C
if a relation S is reflexive and circular, then S is an equivalence relation.
D
if a relation S is circular and symmetric, then S is an equivalence relation.
4
GATE CSE 2019
+2
-0.67
Consider the first order predicate formula φ:

∀x[(∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒ ((w = z) ∨ (z = 1)))]

Here 'a|b' denotes that 'a divides b', where a and b are integers.

Consider the following sets:

S1. {1, 2, 3, ..., 100}
S2. Set of all positive integers
S3. Set of all integers

Which of the above sets satisfy φ?
A
S1 and S3
B
S1, S2 and S3
C
S2 and S3
D
S1 and S2
GATE CSE Subjects
EXAM MAP
Medical
NEET