ESSAY MAMBA

Prove any one of the operations is commutative.

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  1. Define the binary operator  @ by:

aa@b=2bb=2b

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (4(4@6)6)@3=3=
  2.  (2(2@7)7)@5=5=
  1. Define the binary operator  ⊕⊕by:

a⊕b=4aba⊕b=4ab

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (2⊕4)⊕3=(2⊕4)⊕3=
  2.  (5⊕7)⊕6=(5⊕7)⊕6=
  1. Define the binary operator  #  by:

aa#b=b= the larger value of aa or bb.;

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (2(2#6)6) # 7=7=
  2.  (4(4#9)9) # 3=3=
  1. Define the binary operator  ⋄⋄by:

a⋄b=4a+4ba⋄b=4a+4b

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (3⋄7)⋄5=(3⋄7)⋄5=
  2.  (2⋄4)⋄6=(2⋄4)⋄6=
  1. Define the binary operator  ⋆⋆by:

a⋆b=a+2ba⋆b=a+2b

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (7⋆4)⋆5=(7⋆4)⋆5=
  2.  (6⋆2)⋆3=(6⋆2)⋆3=
  1. Define the binary operator  ∇∇by:

a∇b=4a∇b=4

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (6∇7)∇4(6∇7)∇4 =
  2.  (2∇5)∇3(2∇5)∇3 =
  1. Define the binary operator  ⊗⊗by:

a⊗b=a2+b+2a⊗b=a2+b+2

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (4⊗6)⊗3=(4⊗6)⊗3=
  2.  (2⊗5)⊗7=(2⊗5)⊗7=
  1. Define the binary operator  □□by:

a□b=a2+b2a□b=a2+b2

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (4□7)□6=(4□7)□6=
  2.  (2□5)□3=(2□5)□3=
  1. Define the binary operator  & by:

aa & b=2a+4b+5b=2a+4b+5

Simplify each of the following.  Do the order of operations (do what is in parentheses first).

  1.  (2(2 & 3)3)  &  44  =
  2.  (5(5 & 7)7)  &  66  =
  1. Define the binary operator ⊗⊗by:

a⊗b=a2+b+2a⊗b=a2+b+2

and @ by:

aa@b=5bb=5b

Find the following.  When simplifying, use the order of operations, that is, do the parentheses first.

(6(6 ⊗⊗ 8)8) @ 7=7=

 

  1. Define the binary operator ⊕⊕by:

a⊕b=2aba⊕b=2ab

and ⋆⋆ by:

a⋆b=a+8ba⋆b=a+8b

Find the following.  When simplifying, use the order of operations, that is, do the parentheses first.

(7(7 ⊕⊕ 5)5) ⋆⋆ 3=3=

 

  1. Define the binary operator ⊕⊕by:

a⊕b=5aba⊕b=5ab

and ⋄⋄ by:

a⋄b=6a+6ba⋄b=6a+6b

Find the following.  When simplifying, use the order of operations, that is, do the parentheses first.

(4(4 ⊕⊕ 3)3) ⋄⋄ 2=2=

 

  1. Consider the following operators:
  1.   🙂 where aa :)b=4a+7b+2b=4a+7b+2
  2.   ∇∇ where a∇b=6a∇b=6
  3.   ⋄⋄ where a⋄b=2a+2ba⋄b=2a+2b
  4.   ⋆⋆ where a⋆b=a+5ba⋆b=a+5b
  5.   ⊕⊕ where a⊕b=3aba⊕b=3ab
  6.   ⊗⊗ where a⊗b=a2+b+8a⊗b=a2+b+8
  7.   □□ where a□b=a2+b2a□b=a2+b2
  8.   @ where aa@b=7bb=7b
  9.   # where aa#b=b=the smaller value of aa or bb
  1. Prove any one of the operations is commutative.
  2. Give a counterexample for one of the operations that is not commutative.
  1. Consider the following operators:
  1.   🙂 where aa :)b=8a+6b+2b=8a+6b+2
  2.   ∇∇ where a∇b=5a∇b=5
  3.   ⋄⋄ where a⋄b=7a+7ba⋄b=7a+7b
  4.   ⋆⋆ where a⋆b=a+2ba⋆b=a+2b
  5.   ⊕⊕ where a⊕b=4aba⊕b=4ab
  6.   ⊗⊗ where a⊗b=a2+b+6a⊗b=a2+b+6
  7.   □□ where a□b=a2+b2a□b=a2+b2
  8.   @ where aa@b=3bb=3b
  9.   # where aa#b=b=the smaller value of aa or bb
  1. Prove any one of the operations is associative.
  2. Give a counterexample for one of the operations that is not assocative.

 

Prove any one of the operations is commutative.
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