Let's analyze each pair of functions step by step. Set \( A = \{, 1\} \). --- ## 1. \( f: A \times A \to \mathbb{R} \), \( f(x, y) = (1-x) \cdot |x-y| \) \( g: A \times A \to \mathbb{R} \), \( g(x, y) = y|x-y| \) Let's compute for all possible values of \( x \) and \( y \): - **Case 1:** \( x = , y = \) - \( f(, ) = (1-)|-| = 1 \cdot = \) - \( g(, ) = |-| = \) - **Case 2:** \( x = , y = 1 \) - \( f(, 1) = (1-)|-1| = 1 \cdot 1 = 1 \) - \( g(, 1) = 1|-1| = 1 \cdot 1 = 1 \) - **Case 3:** \( x = 1, y = \) - \( f(1, ) = (1-1)|1-| = \cdot 1 = \) - \( g(1, ) = |1-| = \cdot 1 = \) - **Case 4:** \( x = 1, y = 1 \) - \( f(1, 1) = (1-1)|1-1| = \cdot = \) - \( g(1, 1) = 1|1-1| = 1 \cdot = \) **Conclusion:** \( f(x, y) = g(x, y) \) for all \( x, y \in A \). **Functions are equal.** --- ## 2. \( f: A \times A \to A \times A \), \( f(x, y) = (1, xy) \) \( g: A \times A \to A \times A \), \( g(x, y) = (1, 1 - |x - y|) \) - \( xy \) is 1 only if both \( x = 1 \) and \( y = 1 \), otherwise . - \( |x - y| \) is if \( x = y \), 1 if \( x \neq y \). - \( 1 - |x - y| \) is 1 if \( x = y \), if \( x \neq y \). Let's check all cases for the second component: - \( x = y \): \( xy = x \), \( 1 - |x-y| = 1 \), so if \( x = 1 \): \( xy = 1 \), \( 1 - |x-y| = 1 \), if \( x = \): \( xy = \), \( 1 - |x-y| = 1 \). Not equal. - \( x \neq y \): \( xy = \), \( 1 - |x-y| = \). So, for \( (x, y) = (, ) \), \( f(, ) = (1, ) \), \( g(, ) = (1, 1) \). Not equal. **Conclusion:** **Functions are not equal.** --- ## 3. \( f: A \times A \to \mathbb{Z} \), \( f(x, y) = |2x - 1| - y \) \( g: A \times A \to \mathbb{Z} \), \( g(x, y) = (1-x)(1-y) \) Let's compute for all cases: - \( x = , y = \): \( f(,) = |2 \cdot - 1| - = | - 1| - = 1- = 1 \) \( g(,) = (1-)(1-) = 1 \cdot 1 = 1 \) - \( x = , y = 1 \): \( f(,1) = | - 1| - 1 = 1 - 1 = \) \( g(,1) = (1-)(1-1) = 1 \cdot = \) - \( x = 1, y = \): \( f(1,) = |2 \cdot 1 - 1| - = |2-1| - = 1- = 1 \) \( g(1,) = (1-1)(1-) = \cdot 1 = \) - \( x = 1, y = 1 \): \( f(1,1) = |2-1| - 1 = 1-1 = \) \( g(1,1) = (1-1)(1-1) = \cdot = \) So, the functions differ for \( (1, ) \). **Conclusion:** **Functions are not equal.** --- ## 4. \( f: A \to A \), \( f(x) = \sqrt{x} \) \( g: A \to A \), \( g(x) = x^3 \) Values for \( x = , 1 \): - \( x = \): \( f() = \sqrt{} = \), \( g() = ^3 = \) - \( x = 1 \): \( f(1) = \sqrt{1} = 1 \), \( g(1) = 1^3 = 1 \) **Conclusion:** **Functions are equal.** --- # **Final Answers** 1. **Equal** 2. **Not Equal** 3. **Not Equal** 4. **Equal**