Answer
A) The statement universally True.
B) The statement universally False.
C) The statement universally True.
D) The statement universally False.
E) The statement universally True.
F) The statement universally True.
G) The statement universally False.
H) The statement universally True.
I) The statement universally False.
Work Step by Step
A)
∀n means: "for all integers n."
∃m means: "there exists an integer m."
n² < m means "n squared is less than m."
lets now see see why it is true:
1. Consider any arbitrary integer n.
2. Square n to get n².
3. Since n² is always non-negative (0 or greater) for any integer n, any
integer m greater than n² would satisfy the condition n² < m.
4. Because we can always find an integer m greater than n² for any n, the statement holds true for all integers n.
B)
∃n: This signifies "there exists an integer n."
∀m: This indicates "for all integers m."
n < m²: This represents the condition "n is less than the square of m."
lets see why it is false:
1. Consider any integer n.
2. Regardless of the value of n, there will always be at least one integer m that would make the statement false.
3. If n is positive, any integer m with a value of 0 or a negative integer would violate the condition (n would not be less than m²).
4. If n is negative, any integer m with a positive square root (i.e., any positive integer m) would violate the condition (n would not be less than m²) hence it is false.
C)
lets breakdown the statement:
∀n= "for all integers n."
∃m="there exists an integer m."
n + m = 0 ="the sum of n and m is equal to 0."
1. Consider any integer n.
2. To find an integer m that satisfies the equation, simply take the negative of n and assign it to m.
3. This means m = -n.
4. Substituting this value in the equation:
n + (-n) = 0
5. This equation holds true regardless of the value of n, as n - n always equals 0. Therefore, since we can always find an integer m (specifically, m = -n) for any integer n that satisfies the equation, the statement is universally true for all integers in the domain and hence true.
D)
∃n: This signifies "there exists an integer n."
∀m: This indicates "for all integers m."
nm = m: This represents the condition "the product of n and m is equal to m."
1. For any non-zero integer n:
If m is positive, nm will be greater than m (positive multiplied by positive is positive).
If m is negative, nm will be negative (positive multiplied by negative is negative), still not equal to m.
2. For n = 0:
The equation holds true only for m = 0 (0m = 0). However, the statement claims that this is true for all integers m, which is not the case.
Therefore, there is NO single integer n that satisfies the condition for all integers m. This makes the statement false.
E)
∃n: This signifies "there exists an integer n."
∃m: This indicates "there exists an integer m."
n² + m² = 5: This represents the condition "the sum of the squares of n and m is equal to 5."
1. Consider the following pair:
n = 1 m = 2
2. Substitute these values into the equation:
1² + 2² = 1 + 4 = 5 which is true
Therefore, we have found a pair of integers (n = 1, m = 2) that satisfies the condition. Since we only need one such pair to prove the statement's truth, we can conclude that:
F)
∃n: This signifies "there exists an integer n."
∃m: This indicates "there exists an integer m."
n² + m² = 6: This represents the condition "the sum of the squares of n and m is equal to 6."
1. Consider the following pair:
n = 1 m = 1
2. Substitute these values into the equation:
1² + 1² = 1 + 1 = 2
3. This equation does not hold true.
However, this doesn't mean the statement is false yet. We can explore other possibilities:
1. Consider the following pair:
n = 2 m = 0
2. Substitute these values into the equation:
2² + 0² = 4 + 0 = 4
3. This equation also doesn't hold true.
We need to continue trying different pairs until we find one that satisfies the equation:
1. Consider the following pair:
n = 2 m = 1
2. Substitute these values into the equation:
2² + 1² = 4 + 1 = 5
3. This equation finally holds true.
Therefore, we have found a pair of integers (n = 2, m = 1) that satisfies the condition. Since we only need one such pair to prove the statement's truth, we can conclude that:
∃n∃m(n² + m² = 6) is true.
G)
The statement ∃n∃m(n + m = 4 ∧ n - m = 1) translates to "There exist integers n and m such that n + m = 4 and n - m = 1."
Let's evaluate the truth value of this statement:
To show that the statement is true, we need to find specific integers n and m that satisfy both equations simultaneously.
Let's solve the system of equations:
n + m = 4
n - m = 1
Adding equation (1) and equation (2), we get:
(n + m) + (n - m) = 4 + 1
=> 2n = 5
=> n = 5/2
Substituting n = 5/2 into equation (1), we find:
5/2 + m = 4
=> m = 4 - 5/2
=> m = 3/2
Now, n = 5/2 and m = 3/2 are not integers. Therefore, the statement ∃n∃m(n + m = 4 ∧ n - m = 1) is False
H)
∃n: This signifies "there exists an integer n."
∃m: This indicates "there exists an integer m."
n + m = 4 ∧ n − m = 2: This represents the conjunction (and) of two conditions:
The first condition: the sum of n and m is equal to 4.
The second condition: the difference between n and m is equal to 2.
We cannot directly solve for both n and m from the given equations alone. However, we can manipulate them to make finding a solution easier.
Add the two equations together:
(n + m) + (n - m) = 4 + 2
2n = 6
n = 6 / 2
n = 3
Substitute the value of n (3) back into either of the original equations:
n + m = 4
3 + m = 4
m = 4 - 3
m = 1
We found a possible solution: n = 3 and m = 1. Let's verify if it satisfies both conditions:
n + m = 3 + 1 = 4 (True)
n - m = 3 - 1 = 2 (True) which make it true.
I)
∀n: This signifies "for all integers n."
∀m: This indicates "for all integers m."
∃p: This signifies "there exists an integer p such that..."
p = (m + n)/2: This represents the condition "p is equal to the average of m and n."
The statement claims that no matter what integers you choose for n and m, there will always exist another integer p that is exactly the average of n and m. However, this is not true because:
If the average of m and n is not an integer:
If (m + n) is odd, then their average will be a non-integer number (e.g., m = 3, n = 5, (m + n)/2 = 4/2 = 2). In this case, there is no integer p that can satisfy the equation p = (m + n)/2.
Therefore, the statement is False
