Anonymous
Feb 11
46. For any two subsets A and B of a universal set U, prove that: a. A∆ B=B∆ A. b. A∆ B=( A∪B) –(A ∩B). c. A∆ ϕ=A. d. A∆ A=ϕ�?
46. For any two subsets A and B of a universal set U, prove that:
a. A∆ B=B∆ A.
b. A∆ B=( A∪B) –(A ∩B).
c. A∆ ϕ=A.
d. A∆ A=ϕ�
a. A∆ B=B∆ A.
b. A∆ B=( A∪B) –(A ∩B).
c. A∆ ϕ=A.
d. A∆ A=ϕ�
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Feb 11
a. To prove that A∆B = B∆A, we need to show that the symmetric difference operation is commutative.
Let x be an element in A∆B. This means that x is in A or B, but not both.
If x is in A, then x is not in B. Therefore, x is in B∆A.
If x is in B, then x is not in A. Therefore, x is in A∆B.
Since x is in A∆B if and only if x is in B∆A, we can conclude that A∆B = B∆A.
b. To prove that A∆B = (A∪B) – (A∩B), we need to show that the symmetric difference operation is equivalent to the set difference operation.
Let x be an element in A∆B. This means that x is in A or B, but not both.
If x is in A, then x is in A∪B. However, if x is also in A∩B, then it is in both A and B, which contradicts the definition of symmetric difference. Therefore, x is not in A∩B.
If x is in B, then x is in A∪B. However, if x is also in A∩B, then it is in both A and B, which contradicts the definition of symmetric difference. Therefore, x is not in A∩B.
Therefore, x is in (A∪B) – (A∩B).
Now, let x be an element in (A∪B) – (A∩B). This means that x is in A∪B, but not in A∩B.
If x is in A, then x is not in B. Therefore, x is in A∆B.
If x is in B, then x is not in A. Therefore, x is in A∆B.
Therefore, x is in A∆B.
Since x is in A∆B if and only if x is in (A∪B) – (A∩B), we can conclude that A∆B = (A∪B) – (A∩B).
c. To prove that A∆ϕ = A, we need to show that the symmetric difference operation with an empty set results in the original set.
Let x be an element in A∆ϕ. This means that x is in A or ϕ, but not both.
If x is in A, then x is not in ϕ. Therefore, x is in A.
If x is in ϕ, then x is not in A. Therefore, x is not in A.
Therefore, x is in A.
Since x is in A∆ϕ if and only if x is in A, we can conclude that A∆ϕ = A.
d. To prove that A∆A = ϕ, we need to show that the symmetric difference operation with the same set results in an empty set.
Let x be an element in A∆A. This means that x is in A or A, but not both.
Since x is in A, it cannot be not in A. Therefore, there is no element x that satisfies the condition for A∆A.
Therefore, A∆A is an empty set.
Therefore, A∆A = ϕ.
Let x be an element in A∆B. This means that x is in A or B, but not both.
If x is in A, then x is not in B. Therefore, x is in B∆A.
If x is in B, then x is not in A. Therefore, x is in A∆B.
Since x is in A∆B if and only if x is in B∆A, we can conclude that A∆B = B∆A.
b. To prove that A∆B = (A∪B) – (A∩B), we need to show that the symmetric difference operation is equivalent to the set difference operation.
Let x be an element in A∆B. This means that x is in A or B, but not both.
If x is in A, then x is in A∪B. However, if x is also in A∩B, then it is in both A and B, which contradicts the definition of symmetric difference. Therefore, x is not in A∩B.
If x is in B, then x is in A∪B. However, if x is also in A∩B, then it is in both A and B, which contradicts the definition of symmetric difference. Therefore, x is not in A∩B.
Therefore, x is in (A∪B) – (A∩B).
Now, let x be an element in (A∪B) – (A∩B). This means that x is in A∪B, but not in A∩B.
If x is in A, then x is not in B. Therefore, x is in A∆B.
If x is in B, then x is not in A. Therefore, x is in A∆B.
Therefore, x is in A∆B.
Since x is in A∆B if and only if x is in (A∪B) – (A∩B), we can conclude that A∆B = (A∪B) – (A∩B).
c. To prove that A∆ϕ = A, we need to show that the symmetric difference operation with an empty set results in the original set.
Let x be an element in A∆ϕ. This means that x is in A or ϕ, but not both.
If x is in A, then x is not in ϕ. Therefore, x is in A.
If x is in ϕ, then x is not in A. Therefore, x is not in A.
Therefore, x is in A.
Since x is in A∆ϕ if and only if x is in A, we can conclude that A∆ϕ = A.
d. To prove that A∆A = ϕ, we need to show that the symmetric difference operation with the same set results in an empty set.
Let x be an element in A∆A. This means that x is in A or A, but not both.
Since x is in A, it cannot be not in A. Therefore, there is no element x that satisfies the condition for A∆A.
Therefore, A∆A is an empty set.
Therefore, A∆A = ϕ.
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MOJO SAID
Feb 11
51. Prove the following are valid arguments by giving formal proof.
c. If the rain does not come, the crops are ruined and the people will starve. The
crops are not ruined or the people will not starve. Therefore, the rain comes.
d. If the team is late, then it cannot play the game. If the referee is here then the
team can play the game. The team is late. Therefore, the referee is not here. �
c. If the rain does not come, the crops are ruined and the people will starve. The
crops are not ruined or the people will not starve. Therefore, the rain comes.
d. If the team is late, then it cannot play the game. If the referee is here then the
team can play the game. The team is late. Therefore, the referee is not here. �
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