Only four propositions, How many models | UGC NET

UGC NET 2018 : 

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) 
A) 8
B) 7
C) 15
D) 16
Solution:
We know there are total 24 = 16 cases.
As shown in below truth table, it won’t satisfy the condition when A = B = C = D = 0.

S.No

A

B

C

D

(⌐ A ​ ⌐B ​ ⌐C ​ ⌐ ​D)

1

0

0

0

0

0

2

0

0

0

1

1

3

0

0

1

0

1

4

0

0

1

1

1

5

0

1

0

0

1

6

0

1

0

1

1

7

0

1

1

0

1

8

0

1

1

1

1

9

1

0

0

0

1

10

1

0

0

1

1

11

1

0

1

0

1

12

1

0

1

1

1

13

1

1

0

0

1

14

1

1

0

1

1

15

1

1

1

0

1

16

1

1

1

1

1

So, from the given sentence false(0) occurs only if A, B, C and D are false(0) which occurs 1 time.
Required number of models = 16 – 1 = 15.