Question

A superhero is allowed access to a secure Avengers facility if he/she meets any of the following criteria:

• The superhero has Avengers’ membership and possesses a high-security clearance badge
                                            OR
• The superhero does not have Avengers membership but holds a special permit issued by S.H.I.E.L.D. along with a high-security clearance badge 
                                            OR 
• The superhero is not a recognised ally but holds a special permit issued by S.H.I.E.L.D. along with a high-security clearance badge

The inputs are:

INPUTS
A	Superhero has Avengers membership.
S	Superhero holds a special permit issued by S.H.I.E.L.D.
C	Superhero possesses a high-security clearance badge
L	Superhero is a recognised ally

(In all the above cases, 1 indicates YES and 0 indicates NO) 

Output: X - Denotes allowed access [1 indicates YES and 0 indicates NO in all cases]

Draw the truth table for the inputs and outputs given above. Write the POS expression for X (A, S, C, L).

Answer 1

A superhero is granted access (X = 1) to the Avengers facility if any one of the following conditions is fulfilled:

1. The superhero is an Avengers member (A = 1) and has a high-level security clearance badge (C = 1).
2. The superhero is not an Avengers member (A = 0), but has a special authorization from S.H.I.E.L.D. (S = 1) and also carries a high-security clearance badge (C = 1).
3. The superhero is not a recognised ally (L = 0), but holds a special permit issued by S.H.I.E.L.D. (S = 1) along with a high-security clearance badge (C = 1).

Constructing truth table:

A	S	C	L	X(Access Allowed)
0	0	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	1	1	0
0	1	0	0	0
0	1	0	1	0
0	1	1	0	1
0	1	1	1	1
1	0	0	0	0
1	0	0	1	0
1	0	1	0	1
1	0	1	1	1
1	1	0	0	0
1	1	0	1	0
1	1	1	0	1
1	1	1	1	1

POS (Product of Sums) Expression: is derived from the rows where X =v0 (denied access). These rows are:

(A = 0, S = 0, C = 0, L = 0) (A = 0, S = 0, C = 0, L = 1) (A = 0, S = 0, C = 1, L = 0) (A = 0, S = 0, C = 1, L = 1) (A = 0, S = 1, C = 0, L = 0) (A = 0, S = 1, C = 0, L = 1) (A = 1, S = 0, C = 0, L = 0) (A = 1, S = 0, C = 0, L = 1) (A = 1, S = 1, C = 0, L = 0) (A = 1, S = 1, C = 0, L = 1)

Final : POS expression for X(A, S, C, L) is:

(A + S + C) • (A + S + L) • (A + C + L) • (S + C + L)