1
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
2
GATE CSE 2018
+2
-0.6
Let N be the set of natural numbers. Consider the following sets.

$$\,\,\,\,\,\,\,\,$$ $$P:$$ Set of Rational numbers (positive and negative)
$$\,\,\,\,\,\,\,\,$$ $$Q:$$ Set of functions from $$\left\{ {0,1} \right\}$$ to $$N$$
$$\,\,\,\,\,\,\,\,$$ $$R:$$ Set of functions from $$N$$ to $$\left\{ {0,1} \right\}$$
$$\,\,\,\,\,\,\,\,$$ $$S:$$ Set of finite subsets of $$N.$$

Which of the sets above are countable?

A
$$Q$$ and $$S$$ only
B
$$P$$ and $$S$$ only
C
$$P$$ and $$R$$ only
D
$$P, Q$$ and $$S$$ only
3
GATE CSE 2016 Set 2
+2
-0.6
A binary relation $$R$$ on $$N \times N$$ is defined as follows: $$(a,b)R(c,d)$$ if $$a \le c$$ or $$b \le d.$$ Consider the following propositions:

$$P:$$ $$R$$ is reflexive
$$Q:$$ $$R$$ is transitive

Which one of the following statements is TRUE?

A
Both $$P$$ and $$Q$$ are true
B
$$P$$ is true and $$Q$$ is false
C
$$P$$ is false and $$Q$$ is true
D
Both $$P$$ and $$Q$$ are false
4
GATE CSE 2016 Set 2
+2
-0.6
Consider a set $$U$$ of $$23$$ different compounds in a Chemistry lab. There is a subset $$S$$ of $$U$$ of $$9$$ compounds, each of which reacts with exactly $$3$$ compounds of $$U.$$ Consider the following statements:

$$\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,$$ Each compound in $$U \ S$$ reacts with an odd number of compounds.
$$\,\,\,\,\,{\rm I}{\rm I}.\,\,\,\,\,$$ At least one compound in $$U \ S$$ reacts with an odd number of compounds.
$$\,\,\,{\rm I}{\rm I}{\rm I}.\,\,\,\,\,$$ Each compound in $$U \ S$$ reacts with an even number of compounds.

Which one of the above statements is ALWAYS TRUE?

A
Only $${\rm I}$$
B
Only $${\rm II}$$
C
Only $${\rm III}$$
D
None
GATE CSE Subjects
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Medical
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