Question

Define the binary operator ∇ by: a ∇ b = 4 Simplify each of the following. Do the order of operations (do what is in parentheses first). ( 4 ∇ 3 ) ∇ 6 = ( 7 ∇ 2 ) ∇ 5 =

Answer 1

When the binary operator ∇ is defined as always producing the result 4, both expressions (4 ∇ 3) ∇ 6 and (7 ∇ 2) ∇ 5 simplify to 4, regardless of the numbers involved in the operations.

Step-by-step explanation:

The binary operator ∧ is defined such that for any numbers a and b, the result of a ∧ b is always 4. Using this definition, we simplify the given expressions, starting with the operations inside the parentheses first, as instructed by the order of operations (PEMDAS/BODMAS).

The expression (4 ∧ 3) ∧ 6 simplifies to 4 ∧ 6 because 4 ∧ 3 equals 4. Then, using the definition of the binary operator again, 4 ∧ 6 equals 4.

The expression (7 ∧ 2) ∧ 5 simplifies to 4 ∧ 5 because 7 ∧ 2 equals 4. Again, 4 ∧ 5 equals 4 based on the definition of the binary operator ∧.

Therefore, both expressions simplify to the number 4. There is no need to consider the numerical values within the operators because the result is predefined as 4 regardless of the numbers.

Answer 2

Given that a ∇ b = 4, it doesn't matter what values you choose for a and b, the result of the ∇ operation is always 4.

So, unless your question is missing some additional details, we have

(4 ∇ 3) ∇ 6 = 4 ∇ 6 = 4

and

(7 ∇ 2) ∇ 5 = 4 ∇ 5 = 4