Final answer:
The operations a⊖b = a – |b| and a⊕b = (ab)² are given. The false statements are: a⊖b has a commutative property and a⊖b has an associative property. The true statements are: a⊕b has a commutative property and a⊕b has an associative property.
Step-by-step explanation:
The operations given are:
A) a ⊖ b has a commutative property: False
To check for commutative property, we need to see if a⊖b is equal to b⊖a. If we consider a=2 and b=3, then a⊖b = 2 - |3| = -1, while b⊖a = 3 - |2| = 1. Since -1 is not equal to 1, the operation doesn't have a commutative property.
C) a ⊕ b has a commutative property: True
To check for commutative property, we need to see if a⊕b is equal to b⊕a. If we consider a=2 and b=3, then a⊕b = (2*3)² = 36, while b⊕a = (3*2)² = 36. Since 36 is equal to 36, the operation has a commutative property.
B) a ⊖ b has an associative property: False
To check for associative property, we need to see if a⊖(b⊖c) is equal to (a⊖b)⊖c. If we consider a=2, b=3, and c=4, then a⊖(b⊖c) = 2 - |(3-4)| = -3, while (a⊖b)⊖c = (2-|3|)⊖4 = -1. Since -3 is not equal to -1, the operation doesn't have an associative property.
D) a ⊕ b has an associative property: True
To check for associative property, we need to see if a⊕(b⊕c) is equal to (a⊕b)⊕c. If we consider a=2, b=3, and c=4, then a⊕(b⊕c) = (2*(3*4))² = 1152, while (a⊕b)⊕c = ((2*3)*4)² = 1152. Since 1152 is equal to 1152, the operation has an associative property.