Let's break down the problem using the properties of ordinals in set theory. First, let's understand the expressions involved: 1. is the first infinite ordinal. 2. + 1 is the ordinal that comes immediately after . 3. * 2 represents the ordinal sum of two 's, which is the same as + . 4. ( + 1) * 2 means we are taking the ordinal ( + 1) and adding it to itself, which is ( + 1) + ( + 1). 5. ( * 2) + (1 * 2) means we are taking * 2 and adding it to 1 * 2, which simplifies to ( + ) + 2. Now, let's calculate each expression separately: 1. ( + 1) * 2: - This is ( + 1) + ( + 1). - The first ( + 1) gives us followed by 1. - Adding another ( + 1) means we append another sequence of followed by 1 to the first sequence. - Hence, we get , 1, , 1. 2. ( * 2) + (1 * 2): - * 2 is + . - 1 * 2 is 1 + 1, which is 2. - Adding these together, ( + ) + 2, means we have two sequences of followed by 2. - Hence, we get , , 1, 1. Clearly, ( + 1) * 2 results in a different order and structure compared to ( * 2) + (1 * 2). Therefore, we can see that: ( + 1) * 2 = ( * 2) + (1 * 2) The final answer is: ( + 1) * 2 = ( * 2) + (1 * 2)