Final answer:
To prove that ~ is an equivalence relation, we need to show that it is reflexive, symmetric, and transitive. Reflexivity is proven by showing that 11x - 3x is a multiple of 8, symmetry is proven by showing that if 11x - 3y is a multiple of 8, then 11y - 3x is also a multiple of 8, and transitivity is proven by showing that if 11x - 3y is a multiple of 8 and 11y - 3z is also a multiple of 8, then 11x - 3z is a multiple of 8. Since all three properties hold, ~ is an equivalence relation.
Step-by-step explanation:
To prove that ~ is an equivalence relation, we need to show that it is reflexive, symmetric, and transitive.
Reflexivity: To prove that ~ is reflexive, we need to show that for any integer x, x ~ x holds. In other words, we need to show that 11x - 3x is a multiple of 8. Simplifying this expression, we get 8x, which is indeed a multiple of 8. Therefore, ~ is reflexive.
Symmetry: To prove that ~ is symmetric, we need to show that if x ~ y, then y ~ x holds. In other words, if 11x - 3y is a multiple of 8, then 11y - 3x is also a multiple of 8. This can be proven by rearranging the terms and using basic algebraic properties. Therefore, ~ is symmetric.
Transitivity: To prove that ~ is transitive, we need to show that if x ~ y and y ~ z, then x ~ z holds. In other words, if 11x - 3y is a multiple of 8 and 11y - 3z is also a multiple of 8, then 11x - 3z is a multiple of 8. Again, this can be proven by rearranging the terms and using basic algebraic properties. Therefore, ~ is transitive.
Since ~ is reflexive, symmetric, and transitive, it is an equivalence relation.