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Chapter 2: Problem 1

Suppose a family contains two children of different ages, and we are interested in the gender of these children. Let \(F\) denote that a child is female and \(M\) that the child is male and let a pair such as FM denote that the older child is female and the younger is male. There are four points in the set \(S\) of possible observations: $$S=\\{F F, F M, M F, M M\\}$$ Let \(A\) denote the subset of possibilities containing no males; \(B\), the subset containing two males; and \(C,\) the subset containing at least one male. List the elements of \(A, B, C, A \cap B, A \cup B, A \cap C\) \(A \cup C, B \cap C, B \cup C,\) and \(C \cap \bar{B}\)

Short Answer

Expert verified
A={FF}; B={MM}; C={FM, MF, MM}; A鈭〣=鈭�; A鈭狟={FF, MM}; A鈭〤=鈭�; A鈭狢=S; B鈭〤={MM}; B鈭狢={FM, MF, MM}; C鈭〣虆={FM, MF}.

Step by step solution

01

Identify the Sets

First, identify the given sets:- Set \(S\): the set of all possible observations is \( \{FF, FM, MF, MM\} \).- Set \(A\): contains no males, hence \(A = \{FF\}\).- Set \(B\): contains two males, hence \(B = \{MM\}\).- Set \(C\): contains at least one male, hence \(C = \{FM, MF, MM\}\).
02

Find Intersections and Unions

Now we'll find the intersections and unions of the identified sets:- Intersection \(A \cap B\): There are no outcomes that are both set \(A\) and set \(B\), so \(A \cap B = \{\}\).- Union \(A \cup B\): Combine \(A\) and \(B\) to get \(A \cup B = \{FF, MM\}\).- Intersection \(A \cap C\): There are no outcomes that are both in \(A\) and \(C\), so \(A \cap C = \{\}\).
03

Calculate Additional Intersections and Unions

Let's calculate the additional intersections and unions:- Union \(A \cup C\): Combine \(A\) and \(C\) to get \(A \cup C = \{FF, FM, MF, MM\}\), which is the entire set \(S\).- Intersection \(B \cap C\): They have only one outcome in common, \(B \cap C = \{MM\}\).- Union \(B \cup C\): Combine \(B\) and \(C\) to get \(B \cup C = \{FM, MF, MM\}\).
04

Calculate Complement and Final Intersection

Finally, find the complement of \(B\) and the intersection with \(C\):- Complement \(\bar{B}\) of \(B\): All elements in \(S\) not in \(B\), \(\bar{B} = \{FF, FM, MF\}\).- Intersection \(C \cap \bar{B}\): This is \(\{FM, MF\}\) as they are common in \(C\) and \(\bar{B}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set Theory
Set theory is an essential part of mathematics that deals with the study of collections of objects. In probability theory, sets are used to represent different possible outcomes or events. For our exercise, the set \(S = \{FF, FM, MF, MM\}\) represents all possible combinations of two children's genders.
To understand the exercise's solution, we should grasp some basic operations of set theory:
  • Union \( (A \cup B) \): This operation combines all elements from sets \(A\) and \(B\). For example, \(A \cup B = \{FF, MM\}\) means we include all combinations where there are no males and where there are two males.
  • Intersection \( (A \cap B) \): This operation finds common elements between sets \(A\) and \(B\). For instance, \(A \cap B = \{\} \) indicates there are no common elements between these sets.
  • Complement \( (\bar{B}) \): Represents all elements not in set \(B\). In our example, \(\bar{B} = \{FF, FM, MF\}\).
The correct understanding and application of these operations enable us to analyze and deduce the potential outcomes correctly.
Combinatorics
Combinatorics is the field of mathematics focused on counting, arranging, and combining objects. In our problem, combinatorics helps us to find all possible combinations of two children with respect to their genders. The approach taken involves creating a set of all possible combinations.
  • We denote female as \(F\) and male as \(M\).
  • Considering the order of age, an older and younger child, we compute all permutations - resulting in the set \( \{ FF, FM, MF, MM \} \).
Using combinatorics, we establish how many outcomes adhere to certain rules:
  • The subset \(A = \{FF\}\) signifies only female children.
  • The subset \(B = \{MM\}\) signifies only male children.
  • Subset \(C = \{FM, MF, MM\}\) includes at least one male.
Understanding and arranging these combinations allows us to easily track and predict outcomes in probability problems.
Mathematical Logic
Mathematical logic enables us to use reasoning to solve problems. In our exercise, logic helps us understand and manipulate sets to find intersections and unions of particular criteria.
  • Logical symbols and operations, such as "and" \((\cap)\), "or" \((\cup)\), "not" or complement \((\bar{B})\), structure our problem-solving approach.
  • For instance, finding \(C \cap \bar{B}\) requires determining elements common in both \(C\) and the complement of \(B\), which are \( \{FM, MF\}\).
Logical reasoning also supports us to avoid misinterpretations: understanding relationships between sets like having "at least one male" or "two males" clarifies the subsets we need.
Introducing concepts like intersection (commonality), union (combination), and complement (opposite) enriches our problem-solving toolkit in probability and beyond.

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