Determine whether the statement

∀x ∈ Z, ∃y, z ∈ Z[x = 5y + 7z] is true or false. Explain.

Translation into English:

For all x which are integers, there exist y and z which also are integers, such that x=5y+7z.

Is the above statement true?

Give it a few tries: triplet means (x,y,z)
(0,7,-5) is a solution
(1,10,-7) is a solution
(2,-1,1) is a solution
(3,2,-1) is a solution
(4,-2,2) is a solution
(5,-6,5) is a solution
(6,-3,3) is a solution
All other integers can therefore be obtained by adjusting values of z.

To determine whether the statement ∀x ∈ Z, ∃y, z ∈ Z[x = 5y + 7z] is true or false, we need to analyze its logical structure.

The statement uses quantifiers: ∀ (for all) and ∃ (exists).

∀x ∈ Z means "for all x in the set of integers Z."
∃y, z ∈ Z means "there exist y and z in the set of integers Z."

The statement itself, [x = 5y + 7z], expresses an equation that needs to be satisfied.

To check if the statement is true or false, we need to consider two conditions:
1. Can we find values of y and z for each integer x that satisfy the equation?
2. Are there any counterexamples where we can't find values of y and z that satisfy the equation?

Let's consider the equation x = 5y + 7z.

For any given integer x, we can transform the equation by rearranging it:
x - 5y = 7z

Now, let's look at it from a modular arithmetic perspective.

We can rewrite the equation as:
x ≡ 5y (mod 7)

This equivalence implies that for each integer x, there exists an integer y such that when multiplied by 5 (mod 7), it is congruent to x (mod 7).

To determine if this equation holds true, we need to check the values of x mod 7 from 0 to 6.

For x = 0 (mod 7), y can be 0, 1, 2, 3, 4, 5, or 6 because multiplying them by 5 (mod 7) would still satisfy the equation.

Similarly, for x = 1, y can be 3, 10, 17, 24, and so on. This pattern repeats every 7 units until x reaches 5 (mod 7).

For x = 5 (mod 7), there is no integer y that satisfies the equation since 5 and 0 (mod 7) are not congruent when multiplied by 5.

Hence, for all other values of x, we can find integers y and z that satisfy x = 5y + 7z.

Therefore, the statement ∀x ∈ Z, ∃y, z ∈ Z[x = 5y + 7z] is TRUE.