Let (Z_{10} , +, x) be the ring of integers modulo 10, and let S be the subset of Z_{10} represented by {0, 2, 4, 6, 8}. Which of the following statements is FALSE?

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∻Solution by Steps∻ ‖ step 1 ⋮ To check for closure under addition (+) in $ S $, we add each element of $ S $ to every other element and verify if the result is also in $ S $ modulo 10. ‖ ‖ step 2 ⋮ Performing the addition modulo 10: $ 0+0, 0+2, 0+4, 0+6, 0+8, 2+2, 2+4, 2+6, 2+8, 4+4, 4+6, 4+8, 6+6, 6+8, 8+8 $. ‖ ‖ step 3 ⋮ Simplifying the results of addition modulo 10, we find that all sums are in $ S $: $ 0, 2, 4, 6, 8, 4, 6, 8, 0, 8, 0, 2, 2, 4, 6 $. ‖ ‖ step 4 ⋮ To check for closure under multiplication (×) in $ S $, we multiply each element of $ S $ by every other element and verify if the result is also in $ S $ modulo 10. ‖ ‖ step 5 ⋮ Performing the multiplication modulo 10: $ 0×0, 0×2, 0×4, 0×6, 0×8, 2×2, 2×4, 2×6, 2×8, 4×4, 4×6, 4×8, 6×6, 6×8, 8×8 $. ‖ ‖ step 6 ⋮ Simplifying the results of multiplication modulo 10, we find that all products are in $ S $: $ 0, 0, 0, 0, 0, 4, 8, 2, 6, 6, 4, 2, 6, 8, 4 $. ‖ ‖ step 7 ⋮ Identify any false statements based on the closure properties of $ S $ under addition and multiplication modulo 10. ‖ ∻Answer∻ ⚹ All given statements are true if they assert that $ S $ is closed under addition and multiplication in $ Z_{10} $. If any statement claims otherwise, it is false. ⚹ ∻Key Concept∻ ⚹Closure Properties in Modular Arithmetic⚹ ∻Explanation∻ ⚹A subset $ S $ of $ Z_{10} $ is closed under addition and multiplication if, when performing these operations on elements of $ S $, the result is always an element of $ S $ modulo 10. In this case, $ S = \{0, 2, 4, 6, 8\} $ is closed under both operations in $ Z_{10} $.⚹◊What is the additive identity in the ring of integers modulo 10?⍭ Generate me a similar question◊

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