ITPEC FE Morning April 2022 Question 22
A paper tape reader used as a computer input device reads a tape with five rows of holes as shown below. A hole punched in the tape indicates logic 1, and no hole indicates logic 0. As each hole pattern passes under the photocells, the pattern is translated into logic signals as a variable: A, B, C, D, or E. A valid pattern on the tape has at least one hole, and an invalid pattern has no hole or all five holes punched. Which of the following is a logical expression that has logic 1 when a valid pattern is being read and logic 0 when an invalid pattern is being read? Here, + represents logical OR, · represents logical AND, and A represents the negation of A in the logic expression. In the figure, • represents a punched hole on the tape.
Explanation
The negation of "all five holes" OR "no hole" — the expression in (d) is 0 for exactly the two invalid patterns and 1 for every other pattern.
- •A · B · C · D · E is 1 only for 11111, all five holes punched.
- •A · B · C · D · E is 1 only for 00000, no hole.
- •Their OR is 1 exactly when the pattern is invalid, and the bar over the whole expression inverts it: 1 for a valid pattern, 0 for an invalid one.
Check: 11111 and 00000 each make one term 1, giving 0; 10010 makes both terms 0, giving 1.
Answer (d)
Why not others:
- •(a) — the bar covers only the first product, so the expression has the form X + X with X = A · B · C · D · E, which is 1 for every pattern, the invalid ones included.
- •(b) — the first factor rightly rejects 00000, but in the second factor E is not negated: for 11111 it gives 0 + 0 + 0 + 0 + 1 = 1, so all five holes pass as valid, while the valid pattern 11110 is rejected. With E it would equal (d) by De Morgan's law.
- •(c) — by De Morgan's law A + B + C + D + E equals A · B · C · D · E, so both terms detect only 00000: the expression is 1 for the no-hole pattern alone, the opposite of what is wanted.
Key rule: To accept everything except a few patterns, write a product term for each rejected pattern, OR the terms and negate the whole sum; De Morgan's law turns that into an equivalent product of sums.
Basic Theory · AI-written, may contain errors
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