03 - Association

 1. Boolean algebra does have some of its own laws, it also simply borrows some from regular algebra. Association involves:

  grouping variables into brackets

  a trick of binary arithmetic that allows easy cancellation of factors

  the fact that the order of the variables does not matter

  the fact that multiplying all at once is the same as multiplying individually

 2. The Associative Law is what allows you to group variables into brackets, and to break brackets apart again



 3. In Boolean algebra, A+B+C could be laid out as (A+B)+C or as A+(B+C)



 4. Consider the following and select the correct equivalent statement: (2+3)+4= 9

  (2 + 3 + 4) * 2 = 9

  2 * 3 * 4 = 9

 5. The associative law states that if the parentheses (brackets) are rearranged in any way, the values of the expressions will also be immediately and always altered.



 6. Associativity is not the same as commutativity, which addresses whether or not the order of two operands changes the result



 7. Formally, a binary operation ? on a set S is called associative if it satisfies the associative law: (x ? y) ? z =

  x ? (y ? z) for all x, y, z in S.

  x + (y + z) for all x, y, z in S.

  xyz for all x, y, z in S.

  x (y z)+ x for all x, y, z in S.

 8. Assuming a binary operation is associative, ((ab)c)d could also be written as:

  Option 2 and 3 are both valid

  Option 1

  All options are valid but the last option 'abcd' makes the most sense because the brackets, in all cases, are unnecessary

  Option 1 and 6 are both valid

 9. Consider the following example and fill in the blanks.
The concatenation of the three strings 
" ", 

can be computed by concatenating the first two strings
 (giving "hello ") 
and appending the third string ("world"), 

or by joining the second and third string (giving " 
and concatenating the first string ("hello") with the 
result. The two methods produce the same result; 
string concatenation is _____________________



  NOT associative


 10. (A + B) + C could also be written as:

  A * (B + C)

  A (B + C)


  A + (B + C)

 11. (A B) C could also be written as:

  A + (BC)

  A (B C)

  A + B + C


 12. Associative Law: (A + B) C = A (B C)



 13. Associative law: (A + B) + C = A + (B + C)



 14. The associative law also states that A + A B = A



 15. The following is demonstrating which law?: A + (B + C) = (A + B) + C = A + B + C

  the OR Associative Law

  the NON Associative Law

  the AND Associative Law


 16. Complete the equation for the AND Associative Law: A(B.C) = (A.B)C =


  C + A * B

  A . B . C

  A + B + C

 17. The Absorptive law is identical to the Associative law >> A + (A.B) = A



 18. The Associative law allows the_________________ from an expression and regrouping of the variables.

  improving of solutions by changing sign (e.g. OR to AND or AND to OR)

  removal of brackets

  doubling up of brackets

  simplification of any use of OR gates

 19. The following is correctly depicting the OR Associative law: A + (B + C) = (A + B) + C = A + B + C



 20. It is useful to remember that in Boolean Algebra the '+' in ordinary arithemetic is similar to the 'OR' and the '*' is similar to an 'AND'.