Category: Combinational Canonical

Select all boolean expressions below that are part of the canonical representation of the Xor boolean function?

a b a ^ b
0 0 0
0 1 1
1 0 1
1 1 0

Note: In the boolean expressions

the not operator is represented by !
the and operator is represented by &

Question

Category: Combinational Canonical

Select all boolean expressions below that are part of the canonical representation of the Xor boolean function?

a	b	a ^ b
0	0	0
0	1	1
1	0	1
1	1	0

Note: In the boolean expressions

the not operator is represented by !
the and operator is represented by &

BlackTom AI · Accepted Answer

The question asks which boolean expressions are part of the canonical representation of the XOR function. In canonical (sum-of-products) form for XOR, the output is 1 exactly when exactly one input is true, i.e., for the minterms !a & b and a & !b.

Option 1: !a & b. This corresponds to the case where a is 0 and b is 1. Since XOR is true when the inputs differ, this minterm is included in the canonical sum for XOR. It matches the pattern of the one-hot true condition with a false and b true.

Option 2: a & !b. This corresponds to the case where a is 1 and b is 0. Like the previous minterm, it represents a scenario where the inputs differ, so it is another valid term in the XOR canonical form.

Option 3: a & b. This represents the case where both inputs are 1. For XOR, the output should be 0 in this scenario, so this term is not part of the canonical representation. Including it would correspond to the AND of both truths, which is not XOR.

Option 4: !a & !b. This corresponds to both inputs being 0. XOR is also 0 in this case, so this term does not belong in the canonical sum of minterms for XOR. It would be part of an expression representing equivalence or negation of XOR, not XOR itself.

Putting it together, the terms that constitute the canonical XOR expression are the two minterms where the inputs differ: !a & b and a & !b. The other two terms, a & b and !a & !b, do not belong to the canonical XOR representation because they cover the cases where the inputs are equal, not different.

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