Question

Let g:Z × Z → Z × Z be defined by g (m, n) = (2m, m-n). (a) Calculate g(3, 5) and g(-1,4). (b) Determine all the preimages of (0, 0). That is, find all (m, n) e Z × Z such that g(m, n) (0, 0) (c) Determine the set of all the preimages of (8,-3) (d) Determine the set of all the preimages of () (e) Is the following proposition true or false? Justify your conclusion. For each (s,t) E Z × Z, there exists an (m, n) E Z × Z such that g(m, n) = (st).

Answer 1

After computing the function g for given pairs, analyzing the preimages for specific outputs, and evaluating a proposition about the existence of preimages, it's determined that the given function maps pairs of integers and sometimes lacks a preimage for particular values, making the proposition false.

Step-by-step explanation:

The function g maps a pair of integers (m, n) into another pair of integers (2m, m-n). Let's address each part of the question step by step.

Part (a)

To calculate g(3, 5), we plug in 3 for m and 5 for n to get (2(3), 3-5) which is (6, -2). To calculate g(-1, 4), we plug in -1 for m and 4 for n to get (2(-1), -1-4) which is (-2, -5).

Part (b)

To find all preimages of (0, 0), we set up the equations 2m = 0 and m-n = 0, which solve to m = 0 and n = 0. Thus, the only preimage of (0, 0) is (0, 0).

Part (c)

The set of all preimages of (8, -3) can be found by setting up the equations 2m = 8 and m-n = -3. Solving for m and n, we get m = 4 and n = 7. Thus, the preimage is (4, 7).

Part (d)

The question part (d) seems to be cut off and does not contain complete information to provide an answer.

Part (e)

The proposition is false. If we replace s and t with specific numbers where s is not a multiple of t, such as (2, 4), there exists no integer values for (m, n) that satisfies 2m = 2 and m-n = 4, since that would require m = 1, but 1-1 does not equal 4. Therefore, it's not possible for each (s, t) to have a corresponding (m, n) such that g(m, n) = (st).

Answer 2

To solve for specific g(m, n) values, we substitute m and n into the function definition. The preimage for (0,0) is (0,0) and for (8,-3) it is (4,7). The proposition is false as it does not hold for all (s,t) pairs.

Step-by-step explanation:

The function g:Z × Z → Z × Z is defined by g(m, n) = (2m, m-n) for integers m and n.

Calculating g(3, 5) and g(-1,4)

To calculate g(3, 5), we substitute m = 3 and n = 5 into the function to get:

g(3, 5) = (2×3, 3-5) = (6, -2).

Similarly, to calculate g(-1,4), we substitute m = -1 and n = 4 into the function to get:

g(-1, 4) = (2×(-1), -1-4) = (-2, -5).

Determining the Preimages

Preimages of (0, 0): We want to find all pairs (m, n) such that g(m, n) = (0, 0). Setting the equations 2m = 0 and m-n = 0, we find that m = 0 and n = 0. Therefore, the only preimage of (0, 0) is (0, 0).

Preimages of (8,-3): To find preimages of (8, -3), we set 2m = 8 and m-n = -3. Solving these equations gives us m = 4 and n = 7. Therefore, the only preimage of (8, -3) is (4, 7).

Evaluating the Proposition

The proposition states For each (s, t) ∈ Z × Z, there exists an (m, n) ∈ Z × Z such that g(m, n) = (st, t). This is false, because if s and t are both even numbers, st would be divisible by 4, but 2m can only be divisible by 2, not necessarily by 4.