Problem 16 - Discrete Mathematics with Applications

Question

Mark each as true or false. $$\\{0\\}=\varnothing$$

Step-by-Step Solution

Answer

The statement \(\{0\} = \varnothing\) is false, as \(\{0\}\) contains one element (0) while \(\varnothing\) contains no elements.

Step 1: Identify the elements in each set

For this step, we will list the elements in both sets: - \(\{0\}\) contains the element 0: it is a set with one element. - \(\varnothing\) is the empty set, which means it contains no elements at all.

Step 2: Compare the elements in the sets

Now that we know the elements in both sets, we can compare them to see if they are equal: - \(\{0\}\) has one element, 0 - \(\varnothing\) has no elements Since the sets have a different number of elements, they cannot be equal.

Step 3: Determine if the statement is true or false

As we have established that \(\{0\}\) and \(\varnothing\) are not equal, we can conclude that the statement: \[\{0\} = \varnothing\] is false.

Key Concepts

Empty Set

In set theory, the concept of the empty set is fundamental. An empty set, denoted by \( \varnothing \), is a set that contains no elements. Think of it as a container with absolutely nothing inside. The one key feature of an empty set is that it remains the same regardless of its form. For instance, whether you describe it as a set within brackets like \( \{ \} \) or use the symbol \( \varnothing \), it stays identical in meaning. You can spot an empty set easily because, whenever you try to list what’s inside, you’ll find there’s nothing to write down. It is important to remember:

• An empty set is the only set with zero elements.
• Sets like \( \{0\} \) are not empty, as they contain elements (in this case, the number 0).

Students often confuse \( \{0\} \)—a set with one element, 0—with \( \varnothing \). Always remember that while the number zero is a placeholder or marker in mathematics, having it inside a set means the set is not empty.

Set Equality

Set equality is a concept that helps us understand when two sets are considered identical. For two sets to be equal, every element of one set must be an element of the other set, and vice versa. This means both sets should contain exactly the same elements. If even one element is different or if the number of elements varies, the sets are not equal. To illustrate:

• Consider the sets \( \{a, b, c\} \) and \( \{c, a, b\} \). Despite the different order, they are equal because they contain the same elements: \(a\), \(b\), and \(c\).
• On the other hand, \( \{0\} \) and \( \varnothing \) are not equal.

Evaluating set equality involves checking each element against the others. If you find any variance in the contents, the sets are unequal.

Elements of a Set

When discussing sets, it's crucial to understand what elements are. Elements are the objects or members contained in a set. For example, in the set \( \{1, 2, 3\} \), the numbers 1, 2, and 3 are elements. The interesting thing about set elements is that:

• The same element cannot appear more than once in a set. If it does, it’s still considered as just one instance.
• Sets can hold numbers, letters, symbols, or even other sets as elements.

The defining property of elements in a set is whether they belong to that set or not. For instance, \(0\) is an element of the set \( \{0\} \), but it's not an element of the empty set \( \varnothing \). The essence of a set is entirely dictated by its elements. Understanding which elements are part of a set is necessary when determining the relationships between sets, like equality.