Decide whether the following statement is true or false and justify your answer.
if n(A) < n(B), then A( B
False
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To determine whether the statement "if n(A) < n(B), then A ⊂ B" is true or false, we need to understand the concepts involved and apply logical reasoning.
First, let's define the terms:
- n(A) represents the number of elements in set A.
- n(B) represents the number of elements in set B.
- A ⊂ B denotes that set A is a proper subset of set B, meaning all elements of A are also elements of B, and B has at least one element not in A.
Now, let's analyze the statement. It states that if the number of elements in set A is less than the number of elements in set B, then A is a proper subset of B.
To determine whether the statement is true or false, we need to consider a counterexample which shows that the statement is false. This means we need to find an example where n(A) < n(B) but A is not a proper subset of B.
Consider the following example:
A = {1, 2, 3}
B = {1, 2, 3, 4}
In this case, n(A) = 3 and n(B) = 4. Clearly, n(A) < n(B).
However, A is not a proper subset of B because both sets have the same elements. A and B are equal.
Therefore, we have found a counterexample that disproves the statement. Hence, the statement "if n(A) < n(B), then A ⊂ B" is false.
In summary, to decide whether the statement is true or false, we analyzed the concepts involved, applied logical reasoning, and found a counterexample that showed the statement to be false.