Insert ⊆ or ⊈ in each blank to make the resulting statement true. {2, 4, 6} ____ {2, 3, 4, 5, 6}

Verified step by step guidance

1. Identify the two sets given: \(A = \{2, 4, 6\}\) and \(B = \{2, 3, 4, 5, 6\}\).
2. Recall the meaning of the symbols: \(\subseteq\) means 'is a subset of' (all elements of the first set are in the second set, possibly equal), and \(\subset\) means 'is a proper subset of' (all elements of the first set are in the second set, but the sets are not equal).
3. Check if every element of set \(A\) is also in set \(B\). Since \(2\), \(4\), and \(6\) are all in \(B\), \(A\) is a subset of \(B\).
4. Determine if \(A\) is equal to \(B\). Since \(B\) has elements \(3\) and \(5\) that are not in \(A\), the sets are not equal.
5. Conclude that the correct symbol to insert is \(\subset\) because \(A\) is a proper subset of \(B\).

Key Concepts

Subset and Proper Subset

A subset is a set where every element of the first set is also in the second set, denoted by '⊆'. A proper subset means the first set is contained within the second set but is not equal to it, denoted by '⊂' or sometimes 's'. Understanding the difference helps determine which symbol correctly expresses the relationship.

Set Notation and Elements

Set notation uses curly braces to list elements, and understanding the elements in each set is crucial. Comparing elements helps identify if one set is contained within another, which is necessary to decide if the subset or proper subset symbol applies.

Symbol Usage in Set Theory

The symbols '⊆' and '⊂' (or 's') represent different types of subset relationships. '⊆' means subset or equal, while '⊂' or 's' means proper subset (strictly contained). Correct symbol usage ensures accurate mathematical statements about set relationships.