Set Theory


Question 1
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 \in 2^X, the inverse of A is the complement of A.
D
For every A \in 2^X, the inverse of A is A.
GATE CSE 2023   Discrete Mathematics
Question 2
Let S be a set of consisting of 10 elements. The number of tuples of the form (A,B) such that A and B are subsets of S, and A\subseteq B is _______
A
49049
B
59049
C
3524
D
854
GATE CSE 2021 SET-2   Discrete Mathematics


Question 3
If A=\{x, y, z\} and B=\{u, v, w, x\}, and the universe is \{s, t, u, v, w, x, y, z\}. Then (A \cup \bar{B}) \cap(A \cap B) is equal to
A
\{u,v,w,x\}
B
\{x\}
C
\{u,v,w,x,y,z\}
D
\{u,v,w\}
ISRO CSE 2020   Discrete Mathematics
Question 3 Explanation: 
NOTE: This question is Excluded for evaluation. Originally all Options are wrong. We have modified one option.

Click here for detail solution by gateoverflow
Question 4
Consider the first order predicate formula:
\forall x[\forall z\; z|x\Rightarrow ((z=x)\vee (z=1))\Rightarrow \exists w(w> x)\wedge (\forall z \; z|w\Rightarrow ((w=z)\vee (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 \varphi ?
A
S1 and S2
B
S1 and S3
C
S2 and S3
D
S1,S2 and S3
GATE CSE 2019   Discrete Mathematics
Question 5
Let U=\{1,2,,...n\}. Let A=\{(x,X)|x\in X,X\subseteq U\}. Consider the following two statements on |A|.

I. |A|=n2^{n-1}
II. |A|=\sum_{k=1}^{n}k\binom{n}{k}

Which of the above statements is/are TRUE?
A
Only I
B
Only II
C
Both I and II
D
Neither I nor II
GATE CSE 2019   Discrete Mathematics




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