Final answer:
An equivalence relation R on the set Z is shown to be reflexive, symmetric, and transitive by verifying that for all x, y, z in Z, the expressions involving R hold true under the operations of addition and multiplication by integers.
Step-by-step explanation:
To show that a relation R is an equivalence relation on the set of integers Z, we need to demonstrate that R is reflexive, symmetric, and transitive.
Reflexive:
For any integer x, we must have xRx. Since 3x + 4x = 7x is divisible by 7, xRx holds for all x in Z, thus R is reflexive.
Symmetric:
If xRy, then 3x + 4y is divisible by 7. For symmetry, we need to show that yRx. Since 3x + 4y is divisible by 7, 3(4y) + 4(3x) is also divisible by 7 because it's just a multiple of 7, proving that yRx and R is symmetric.
Transitive:
If xRy and yRz, then 3x + 4y and 3y + 4z are both divisible by 7. For xRz to hold, 3x + 4z must be divisible by 7. By subtracting 7y (which is divisible by 7) from the sum of 3x + 4y + 3y + 4z, we get 3x + 4z, which is divisible by 7, showing R is transitive.