Expression Evaluation with Custom Symbols

To accurately evaluate the given mathematical expression, we first need to understand the mapping of the custom symbols to their standard arithmetic operators. This problem tests our ability to interpret given rules and apply the correct order of operations (BODMAS/PEMDAS) in a step-by-step manner.

Understanding Symbol Mappings

The question defines the following transformations for the custom symbols into standard mathematical operators:

Step-by-Step Expression Calculation

The given expression is: \( \Delta 2 \oplus 3 \Delta ((4 \otimes 2) \nabla 4) \)

Let's evaluate this expression by replacing the symbols with their corresponding operators and following the order of operations.

• Step 1: Replace all custom symbols with their defined standard operators.

The expression \( \Delta 2 \oplus 3 \Delta ((4 \otimes 2) \nabla 4) \) becomes:

\( 2 - 3 + ((4 \div 2) \times 4) \)

• Step 2: Apply the order of operations (BODMAS/PEMDAS). We start with the operations inside the innermost parentheses.

Calculate the term \( (4 \div 2) \):

\( 4 \div 2 = 2 \)

Now, substitute this value back into the expression:

\( 2 - 3 + (2 \times 4) \)

• Step 3: Continue with the remaining operation inside the parentheses.

Calculate the term \( (2 \times 4) \):

\( 2 \times 4 = 8 \)

Substitute this value back into the expression:

\( 2 - 3 + 8 \)

• Step 4: Perform addition and subtraction operations from left to right.

First, calculate \( 2 - 3 \):

\( 2 - 3 = -1 \)

The expression simplifies to:

\( -1 + 8 \)

• Step 5: Perform the final addition.

Calculate \( -1 + 8 \):

\( -1 + 8 = 7 \)

Final Expression Value

After performing all the symbol substitutions and arithmetic operations in the correct order, the value of the expression \( \Delta 2 \oplus 3 \Delta ((4 \otimes 2) \nabla 4) \) is 7.