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).
(4(4@6)6)@3=3=
(2(2@7)7)@5=5=
11. 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).
(2⊕4)⊕3=(2⊕4)⊕3=
(5⊕7)⊕6=(5⊕7)⊕6=
12. 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).
(2(2#6)6) # 7=7=
(4(4#9)9) # 3=3=
13. 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).
(3⋄7)⋄5=(3⋄7)⋄5=
(2⋄4)⋄6=(2⋄4)⋄6=
14. 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).
(7⋆4)⋆5=(7⋆4)⋆5=
(6⋆2)⋆3=(6⋆2)⋆3=
15. 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).
(6∇7)∇4(6∇7)∇4 =
(2∇5)∇3(2∇5)∇3 =
16. 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).
(4⊗6)⊗3=(4⊗6)⊗3=
(2⊗5)⊗7=(2⊗5)⊗7=
17. 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).
(4□7)□6=(4□7)□6=
(2□5)□3=(2□5)□3=
18. 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).
(2(2 & 3)3) & 44 =
(5(5 & 7)7) & 66 =
19. 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=
20. 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=
21. 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=
23. Consider the following operators:
🙂 where aa :)b=4a+7b+2b=4a+7b+2
∇∇ where a∇b=6a∇b=6
⋄⋄ where a⋄b=2a+2ba⋄b=2a+2b
⋆⋆ where a⋆b=a+5ba⋆b=a+5b
⊕⊕ where a⊕b=3aba⊕b=3ab
⊗⊗ where a⊗b=a2+b+8a⊗b=a2+b+8
□□ where a□b=a2+b2a□b=a2+b2
@ where aa@b=7bb=7b
# where aa#b=b=the smaller value of aa or bb
Prove any one of the operations is commutative.
Give a counterexample for one of the operations that is not commutative.
25. Consider the following operators:
🙂 where aa :)b=8a+6b+2b=8a+6b+2
∇∇ where a∇b=5a∇b=5
⋄⋄ where a⋄b=7a+7ba⋄b=7a+7b
⋆⋆ where a⋆b=a+2ba⋆b=a+2b
⊕⊕ where a⊕b=4aba⊕b=4ab
⊗⊗ where a⊗b=a2+b+6a⊗b=a2+b+6
□□ where a□b=a2+b2a□b=a2+b2
@ where aa@b=3bb=3b
# where aa#b=b=the smaller value of aa or bb
Prove any one of the operations is associative.
Give a counterexample for one of the operations that is not assocative.