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Sum-of-Products and Product-of-Sums Expressions
22 questions By Tony R. Kuphaldt
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Question 1 of 22
Identify each of these logic gates by name, and complete their respective truth tables:
Reveal answer
Notes:In order to familiarize students with the standard logic gate types, I like to given them practice with identification and truth tables each day. Students need to be able to recognize these logic gate types at a glance, or else they will have difficulty analyzing circuits that use them.
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Question 2 of 22
Identify each of these relay logic functions by name (AND, OR, NOR, etc.) and complete their respective truth tables:
Reveal answer
Notes:In order to familiarize students with standard switch contact configurations, I like to given them practice with identification and truth tables each day. Students need to be able to recognize these ladder logic sub-circuits at a glance, or else they will have difficulty analyzing more complex relay circuits that use them.
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Question 3 of 22
Inspect each of these Boolean expressions, and determine whether each one is a sum of products, or a product of sums:
$$(B+\overline{C}+D)(\overline{A}+B)$$
$$A\overline{B}\overline{C}+\overline{A}BC$$
$$(X+\overline{Y}+\overline{Z})(\overline{Y}+Z)(\overline{X}+Y)$$
$$\overline{M}\overline{N}\overline{O}+MN\overline{O}+M\overline{N}O$$
$$(X+\overline{Y+Z})(\overline{Y+\overline{Z}})$$
$$\overline{ABC}+A\overline{B}C$$
Reveal answer$$(B+\overline{C}+D)(\overline{A}+B)\ POS$$
$$A\overline{B}\overline{C}+\overline{A}BC\ SOP$$
$$(X+\overline{Y}+\overline{Z})(\overline{Y}+Z)(\overline{X}+Y)\ POS$$
$$\overline{M}\overline{N}\overline{O}+MN\overline{O}+M\overline{N}O\ SOP$$
The last two expressions are “trick” questions: while technically being the product of summed variables, and the sum of multiplied (product) variables, respectively, do not follow “standard” POS and SOP forms, because they both have long complementation bars:$$(X+\overline{Y+Z})(\overline{Y+\overline{Z}})$$
$$\overline{ABC}+A\overline{B}C$$
For an expression to properly follow the SOP or POS canonical form, no complementation bar should cover more than one variable!
Notes:Even if your students have never heard of Boolean algebra before, they should still be able to determine which of the first four expressions are SOP and which are POS. If there is any confusion on this point, ask your students to define what “sum” and “product” mean, respectively, and then discuss what it means for an expression to be a product (singular) of sums (multiple), or a sum (singular) of products (multiple).
