If A = {1, 2, 3, 4, 5} , B = {a, b, c, d, e} and C = { -, +, x, @, *} , then which of the following is the not the element of set (A x B x C)?

To solve the problem, we need to determine which option does not belong to the Cartesian product $A \times B \times C$.

1. ### Break Down the Problem

• Identify the sets:

  • Set $A = \{1, 2, 3, 4, 5\}$
  • Set $B = \{a, b, c, d, e\}$
  • Set $C = \{ -, +, \times, @, *\}$

• Understand what the Cartesian product $A \times B \times C$ entails:

  • It's the set of all ordered triples $(a_i, b_j, c_k)$ such that $a_i \in A$, $b_j \in B$, and $c_k \in C$.

2. ### Relevant Concepts

• The Cartesian product $A \times B \times C$ can be calculated as: $A \times B \times C = \{ (x, y, z) \mid x \in A, y \in B, z \in C \}$
• The total number of elements in the Cartesian product is $|A| \times |B| \times |C|$ where $|A| = 5, |B| = 5, |C| = 5$.

3. ### Analysis and Detail

• Calculate the number of elements in $A \times B \times C$: $|A \times B \times C| = 5 \times 5 \times 5 = 125$

• The elements of $A \times B \times C$ are structured as:

  • $(1, a, -), (1, a, +), (1, a, \times), (1, a, @), (1, a, *), \ldots, (5, e, *)$

• Since $A \times B \times C$ will consist solely of ordered triples formed from $A$, $B$, and $C$, any element not fitting this format cannot belong to $A \times B \times C$.

4. ### Verify and Summarize

If provided with options, we can evaluate each option to see if:

• It is in the format of $(x, y, z)$
• Each $x$ is from set $A$
• Each $y$ is from set $B$
• Each $z$ is from set $C$

Final Answer

The element that does not fit the format of the Cartesian product will be the one that does not have an element from each of sets $A$, $B$, and $C$ or does not form a valid ordered triple.

If options are provided, please share them for an accurate identification of the element not in $A \times B \times C$.