Question

In the example 52 Cards, find a simple description for each of the following: (a) SameSuit \(\cap\) Same Value (b) (SameSuit \( \cup\) SameValue)"

Short answer

(a) A single card in the deck. (b) Cards that share a suit or value.

Step-by-step solution

Understanding the Problem

We have a deck of 52 playing cards and are asked to find descriptions for two sets: the intersection of SameSuit and SameValue, and the union of SameSuit and SameValue. 'SameSuit' refers to cards that are of the same suit, and 'SameValue' refers to cards that have the same rank or value.

Finding SameSuit \\cap SameValue

The intersection of 'SameSuit' and 'SameValue' is the set of cards that are simultaneously of the same suit and the same value. This describes an individual card since only one card in a standard deck can have the same suit and value. Thus, the intersection is a set containing all individual cards from the deck.

Finding SameSuit \\cup SameValue

The union of 'SameSuit' and 'SameValue' includes any card that is either the same suit or the same value and possibly both. In this context, the description pertains to any set of cards that are either all of the same suit (like all Hearts) or all of the same value (like all Kings). Thus, this represents all possible groupings where cards share a common suit or value.

Key concepts

Intersection of Sets

The intersection of sets involves identifying elements that are common to both sets. In the context of card games, consider two sets: 'SameSuit', which is a set of all cards with the same suit, and 'SameValue', a set of cards sharing the same rank. The intersection of 'SameSuit' and 'SameValue', denoted by \( \cap \), represents cards that are both the same suit and value.
In a standard 52-card deck, this boils down to individual cards because each card is uniquely defined by its suit and value. For example, within the intersection are cards such as the Ace of Spades or the Queen of Hearts, each standing alone in meeting both criteria.
To understand these intersections better, remember:

• The intersection seeks shared attributes between sets.
• Each element in the intersection must satisfy all conditions required by both sets.
• In a deck, the intersection of 'SameSuit' \( \cap \) 'SameValue' culminates in unique instances - individual cards.

Union of Sets

The union of sets represents a combination of all elements from both sets without repetition. For our card example, 'SameSuit' and 'SameValue' offer a practical illustration when united.
The union \( \cup \) means if a card belongs to 'SameSuit' or 'SameValue', it is included in the result. Imagine drawing all hearts (same suit) or all kings (same value); these sets, though distinct, can both be combined under the union. Hence, a card might be part of one set, or both, and still be accounted for in the union.
Consider these traits when examining unions:

• The union encompasses any element belonging to either set.
• It results in a larger set potentially containing elements from one or both original sets.
• In card terms, it includes every possible grouping by suit or value.

Cardinality of Sets

The cardinality of a set indicates the number of elements within it. In a card scenario, it's often about counting possibilities. For 'SameSuit' \( \cap \) 'SameValue', the cardinality is straightforward: 52, matching solo card counterparts in a deck.
When it comes to 'SameSuit' \( \cup \) 'SameValue', the dynamics shift as combinations grow. Understanding cardinality helps when grouping options by either suit or value.
Some cardinality insights include:

• It reveals the scope or size of a set.
• In 'SameSuit' \( \cup \) 'SameValue', the variety of combinations makes counting more complex.
• Cardinality offers a helpful tool to assess the breadth of a union or the specificity of an intersection.

To get comfortable with cardinality, frequently practice identifying and counting elements of different sets.