Need help with True or False for the following... (I will put what I believe As (T) or (F)) 1. For all natural numbers a, a^2 = 0 (mod 4) or a^2 = 1 (mod 4). (T) 2. For all natural numbers a, if a^2 = 2 (mod 4), then 1 = 2. (T) 3. For all natural numbers a, If a^2 = 2 (mod 4), then 1 != 2. (T) The negation of statement 2, which I believe is "There exists a natural number a such that a^2 ≠ 2 (mod 4) or 1 ≠ 2" (F) The negation of statement 3, which I believe is "There exists a natural number a such that a^2 ≠ 2 (mod 4) and 1 = 2" (F) The converse of statement 2, which I believe is "If 1 = 2, then for all natural numbers a, a^2 = 2 (mod 4)" The converse of statement 3, which I believe is "If 1 ≠ 2, then for all natural numbers a, a^2 = 2 (mod 4)" (F) The contrapositive of statement 2, which is "If 1 ≠ 2, then there exists a natural number a such that a^2 ≠ 2 (mod 4)" The contrapositive of statement 3, which is "If 1 = 2, then there exists a natural number a such that a^2 ≠ 2 (mod 4)" (F) For sets A and B, A = B if and only if A is a subset of B and B is a subset of A.
Need help with True or False for the following... (I will put what I believe As (T) or (F))
1. For all natural numbers a, a^2 = 0 (mod 4) or a^2 = 1 (mod 4). (T)
2. For all natural numbers a, if a^2 = 2 (mod 4), then 1 = 2. (T)
3. For all natural numbers a, If a^2 = 2 (mod 4), then 1 != 2. (T)
The negation of statement 2, which I believe is "There exists a natural number a such that a^2 ≠ 2 (mod 4) or 1 ≠ 2" (F)
The negation of statement 3, which I believe is "There exists a natural number a such that a^2 ≠ 2 (mod 4) and 1 = 2" (F)
The converse of statement 2, which I believe is "If 1 = 2, then for all natural numbers a, a^2 = 2 (mod 4)"
The converse of statement 3, which I believe is "If 1 ≠ 2, then for all natural numbers a, a^2 = 2 (mod 4)" (F)
The contrapositive of statement 2, which is "If 1 ≠ 2, then there exists a natural number a such that a^2 ≠ 2 (mod 4)"
The contrapositive of statement 3, which is "If 1 = 2, then there exists a natural number a such that a^2 ≠ 2 (mod 4)" (F)
For sets A and B, A = B if and only if A is a subset of B and B is a subset of A.
Let's evaluate the provided statements and their negations as well as the converses and contrapositives:
1. Statement: For all natural numbers a, a^2 = 0 (mod 4) or a^2 = 1 (mod 4). (True)
- This is true. Any natural number squared (a^2) will have a remainder of either 0 or 1 when divided by 4.
2. Statement: For all natural numbers a, if a^2 = 2 (mod 4), then 1 = 2. (False)
- This statement is false. It implies that if a^2 is congruent to 2 modulo 4, then 1 must equal 2, which is not true.
3. Statement: For all natural numbers a, if a^2 = 2 (mod 4), then 1 ≠ 2. (True)
- This statement is true. If a^2 is congruent to 2 modulo 4, 1 does not equal 2.
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