Which of the following does not define real field for any a and b?*1 pointa+b = b+aa.(b+c)= a.b +b.ca.b=b.aa +(b+c)= c + (a+b)

To determine which of the provided expressions does not define a real field for any $a$ and $b$, let’s analyze each expression in the context of the field axioms (closure, associativity, commutativity, distributivity, identity elements, and inverses) governing the real numbers.

1. Expression 1: $a + b = b + a$

   • This represents the commutative property of addition, which holds for real numbers.

2. Expression 2: $a \cdot (b + c) = a \cdot b + a \cdot c$

   • This is the distributive property of multiplication over addition, which also holds for real numbers.

3. Expression 3: $a \cdot b = b \cdot a$

   • This states the commutative property of multiplication, which is true for real numbers.

4. Expression 4: $a \cdot b = a \cdot (b + c) = c + (a + b)$

   • This expression is problematic. The left side $a \cdot b$ does not generally equal $a \cdot (b + c)$ and this does not reflect a standard property of fields. Thus, it does not conform to the axioms that define a real field.

Final Answer

The expression that does not define a real field for any $a$ and $b$ is:
$a \cdot b = a \cdot (b + c) = c + (a + b)$ (Expression 4).