Mathematical Logic UGC

Question 1

Match List-I with List-II List-I                          List-II (a) p → q                (i) ¬(q → ¬p) (b) p v q                   (ii) p ∧ ¬q (c) p ∧ q                  (iii) ¬p → q (d) ¬(p → q)            (iv) ¬p v q Choose the correct option from those given below:

• a-ii,b-iii,c-i,d-iv
• a-ii,b-i,c-iii,d-iv
• a-iv,b-i,c-iii,d-ii
• a-iv,b-iii,c-i,d-ii

Question 2

​Consider the vocabulary with only four propositions A,B,C and D. How many models are there for the following sentence ?

(⌐ A ∨​ ⌐B ∨​ ⌐C ∨​ ⌐ ​D) 

• 8
• 7
• 15
• 16

Question 3

Match the List 1 and List 2 and choose the correct answer from the code given below:

• (a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)
• (a)-(ii), (b)-(i),(c)-(iii), (d)-(iv)
• (a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
• (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)

Question 4

How many are there to place 8 indistinguishable balls into four distinguishable bins

• 70

• 165

• 8 C 4

• 8P 4T

Question 5

Which of the following is the principal conjunctive normal form for [(pVq) ∧ ~p → ~q]?

• pV~q
• pVq
• ~pVq
• ~pV~q

Question 6

Consider the following methods: M​ 1 ​ : mean of maximum M​ 2​ : Centre of area M​ 3​ : Height method Which of the following is/are defuzzification method(s)?

• Only M1
• Only M1 and M2
• Only M2 and M3
• All M1, M2 and M3

Question 7

A fuzzy conjunction operator denoted as t(x,y) and a fuzzy disjunction operator denoted as s(x, y) form a dual pair if they satisfy the condition:

• t(x,y)=1−s(x,y)
• t(x,y)=s(1−x,1−y)
• t(x,y)=1−s(1−x,1−y)
• t(x,y)=s(1+x,1+y)

Question 8

Suppose that a connected planar graph has six vertices, each of degrees four. Into how many regions is the plane divided by a planar representation of this graph?

• 6
• 8
• 12
• 10

Question 9

Consider the following two statements with respect to the matrices A_m×n , B_n×m , C_n×n and  D_n×n

Statement 1: tr(AB) = tr(BA) 
Statement 2: tr(CD) = tr(DC) 

where tr() represents the trace of a matrix. Which one of the following holds?

• Statement 1 is correct and Statement 2 is wrong. 

• Statement 1 is wrong and Statement 2 is correct.

• Both Statement 1 and Statement 2 are correct. 

• Both Statement 1 and Statement 2 are wrong.