Can someone teach me how to find all valid minterms of any given truth table?

Explanation:

A minterm is a product of variables, in which each variable appears exactly once, either complemented or not. When a truth table is represented in a sum of products form, there is one minterm for each line in the table for which the output is True (1).

Suppose I have a truth table as follows, with inputs A, B and output Z.

  AB Z

  00 1

  01 0

  10 1

  11 0

When I write the minterms, I use a complemented variable where the input is 0, and an uncomplemented variable where the input is 1. The minterm is expressed as a product of those variables.

Here, there are 2 lines in the table with outputs Z=1. The input conditions AB for those outputs are 00 and 10, so the corresponding minterms are A'B' and AB', where the apostrophe indicates the variable is complemented.

This table is then represented by the sum of products ...

  Z = A'B' + AB'

You have all of the valid minterms when you have one for each line of the table where the output is 1.

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Comment on truth table representation

Truth tables can be represented a number of ways. A nice representation for the purpose of identifying minterms is a list of the output for every possible combination of inputs. We have used that form above.

Sometimes, you may see minterms listed as decimal numbers in the range of 0 to 2^n -1 when there are n input variables. Then, the minterms are developed from the binary equivalents of those decimal numbers.

Example:

Logic with 4 input variables has minterms 7, 8, and 15. The binary equivalents of these numbers are 0111, 1000, 1111. The corresponding minterms (for inputs a, b, c, d) are a'bcd + ab'c'd' + abcd = z. Three numbers are listed, so we know we have all of the valid minterms when we have three corresponding minterms.

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Additional comment

In some cases, outputs will be listed as "don't care." Another set of minterms can be generated for the don't cares. These can be used in the logic minimization process.

The answer is option D (Weeping for hours in bed is how I spent my day, broken after a long, sad week).

In a gerund phrase, the head is a verb that ends in ING. This type of phrase can fulfill all the functions of a noun. For example, in this particular case, "weeping" is the head of the subject.

A participial phrase looks like a verb but it is an adjective since it modifies a noun. In this case, the participial phrase is "broken after..."