Question

Let A be a set and f be a function from A to A. A partition P of A is said to have the substitution property with respect to f if, for any two elements a and b that are together in one block of P, the two elements f(a) and f(b) are also together in one block of P. Let A = {1, 2, 3, 4, 5, 6} and f be a function from A to A such that f(1) = 3, f(2) = 3, f(3) = 2, f(4) = 5, f(5) = 4, f(6) = 4. Then, a. Does P = {123, 456} have the substitution property with respect to f? b. If A is the set of integers and P be a partition of A into even and odd integers; f(a) = a+1, for every a in A. Does P have the substitution property with respect to f? 2. Let A be a set of integers and R1 and R2 be relations on AxA such that the ordered pair (a, b), (c, d) is in R1 if and only if (a - c) = (b - d) and that is in R2 if and only if √(a - c)^2 + (b - d)^2 ≤ 10. What is the geometric meaning of (i) R1 ∪ R2; (ii) R1 ∩ R2; (iii) R1 - R2; (iv) R1ΔR2, Δ is the symmetric difference operation?

          Let A be a set and f be a function from A to A. A partition P of A is said to have the substitution property with respect to f if, for any two elements a and b that are together in one block of P, the two elements f(a) and f(b) are also together in one block of P. Let A = {1, 2, 3, 4, 5, 6} and f be a function from A to A such that f(1) = 3, f(2) = 3, f(3) = 2, f(4) = 5, f(5) = 4, f(6) = 4. Then,
a. Does P = {123, 456} have the substitution property with respect to f?
b. If A is the set of integers and P be a partition of A into even and odd integers; f(a) = a+1, for every a in A. Does P have the substitution property with respect to f?

2. Let A be a set of integers and R1 and R2 be relations on AxA such that the ordered pair (a, b), (c, d) is in R1 if and only if (a - c) = (b - d) and that is in R2 if and only if √(a - c)^2 + (b - d)^2 ≤ 10. What is the geometric meaning of (i) R1 ∪ R2; (ii) R1 ∩ R2; (iii) R1 - R2; (iv) R1ΔR2, Δ is the symmetric difference operation?

Added by Ashley L.

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