Mathematical Proofs and Calculations

  Mathematical Proofs and Calculations 1. Proof of Set Relation Given a function f: A -> B and subsets Y, Z ⊆ B, we aim to prove the relation: f^(-1)(Y ∩ Z) = f^(-1)(Y) ∩ f^(-1)(Z) Proof: To prove this relation, we need to show that an element x is in the left-hand side (LHS) if and only if it is in the right-hand side (RHS). Let x be an element in f^(-1)(Y ∩ Z). This means that f(x) is in Y ∩ Z. By definition of set intersection, f(x) is both in Y and Z. Therefore, x is in both f^(-1)(Y) and f^(-1)(Z), implying x is in the RHS. Conversely, if x is in f^(-1)(Y) ∩ f^(-1)(Z), then x is in both f^(-1)(Y) and f^(-1)(Z). This implies that f(x) is in both Y and Z, hence in Y ∩ Z, leading to x being in the LHS. Therefore, we have shown that f^(-1)(Y ∩ Z) = f^(-1)(Y) ∩ f^(-1)(Z). 2. Calculation of Composite Function Consider the function f: ℝ² -> ℝ² defined by f(x, y) = (zy, x³). To find the formula for fo f (composition of f with itself), we substitute the output of f(x, y) into the function again: f(f(x, y)) = f(zy, x³) = (x³y, (zy)³) = (x³y, z³y³) Therefore, the formula for fo f is fo f(x, y) = (x³y, z³y³). 3. Proof of Equivalence Relation Define relation R on ℤ as xRy if and only if x² + y² is even. To prove R is an equivalence relation, we need to show reflexivity, symmetry, and transitivity. Reflexivity: For any x in ℤ, x² + x² = 2x² is always even. Therefore, xRx for all x in ℤ. Symmetry: If xRy (i.e., x² + y² is even), then yRx since addition is commutative. Therefore, if x² + y² is even, y² + x² is also even. Transitivity: If xRy and yRz, then x² + y² and y² + z² are both even. Adding these two equations gives x² + 2y² + z² = 2k for some integer k. Since the sum of three even numbers is even, xRz holds. Hence, R is reflexive, symmetric, and transitive, satisfying the properties of an equivalence relation. Equivalence Classes: The equivalence class [a] of an integer a contains all integers b such that aRb. In this case, the equivalence class [a] consists of all integers b such that a² + b² is even. These equivalence classes form distinct subsets of ℤ based on the parity of the sum of squares of integers.      

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