Match the following Boolean expressions to their equivalents in digital shorthand. Scored equally weighted.

Question

BlackTom AI · Accepted Answer

First, the task asks us to match the given Boolean expression to its digital shorthand equivalents among several options. The target expression provided is w+xz+(x'y)'. We'll evaluate each option to see which corresponds to that form when interpreted with standard Boolean notation (AND implied by adjacency, OR by +, and NOT by after-apostrophe).

Option 1: "w OR x AND z OR NOT (NOT x AND y)". This reads as w + xz + (x'y)'. This exactly matches the target expression when written with the same grouping and operators, since (NOT x AND y) is x'y, and its complement is (x'y)'. Therefore, this option is a correct match.

Option 2: "w OR x AND z OR NOT (NOT x AND y)". It is the same textual form as Option 1. It would evaluate to the identical Boolean expression w + xz + (x'y)'. However, in many multiple-choice conventions, identical choices are redundant; functionally, this is also a correct match, just duplicated. If the test expects a unique best answer, this would still be considered correct in terms of the Boolean equivalence.

Option 3: "w OR z AND NOT x OR NOT (x AND y)". This reads as w + (z x') + (x y)'. This is structurally different from w + xz + (x'y)' because the first product term is z AND NOT x (zx'), whereas the target uses x AND z (xz) as the middle term, and the last term is NOT (NOT x AND y) rather than NOT (x AND y). The grouping and complement patterns do not match, so this is incorrect.

Option 4: "w OR z AND NOT x AND y". This is w + (z x') y, which simplifies to w + z x' y. That has a different factoring: it uses z and x' together with y, but lacks the standalone xz term and the NOT of (NOT x AND y). This is not equivalent to w + xz + (x'y)'. Therefore, it is incorrect.

Option 5: "w AND NOT z OR NOT (x AND y)". Translated, this is (w z') + (x y)'. This form is significantly different: it replaces the w term with w z' in the product, and the last term is the complement of (x AND y) rather than the complement of (NOT x AND y). It does not correspond to w + xz + (x'y)'. Hence, it is incorrect.

Option 6: "wz'+(xy)'". This is w z' + (xy)'. Here w is ANDed with z', rather than appearing as a separate ORed term, and the second term is NOT (x AND y). This is not equivalent to w + xz + (x'y)' because the w term is not isolated as a pure OR operand; the expression structure is different. Incorrect.

Option 7: "w+x'yz". This reads as w + (x' y z). This is not the same form as w + xz + (x'y)'. It replaces the xz term with x' y z, and omits the NOT (NOT x AND y) portion entirely, so it’s incorrect.

Option 8: "w+x'z+(xy)'". This is w + x'z + (xy)'. This again alters the middle product to x'z instead of xz, and uses (xy)' instead of (x'y)'. The overall structure and complements do not match the target, so it’s incorrect.

Option 9: "w+xz+(x'y)'". This option literally matches the target expression in Option 1, written with identical components: w OR x AND z OR NOT (NOT x AND y). Therefore, this is a correct match.

In summary, the correct matches to the target w + xz + (x'y)' are Option 1 and Option 9 (and, if allowed, Option 2 as an exact duplicate of 1). The other options contain different groupings, complements, or product terms that do not reproduce the target expression.

Match the following Boolean expressions to their equivalents in digital shorthand. Scored equally weighted.

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