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Find out how many students in a class of 140 drink both tea and coffee, given that 80 drink tea, 68 drink coffee, and 19 drink neither.
In a class of 140, 80 Students will drink tea, 68 Students will drink coffee and 19 students will not drink both coffee and tree. How many will drink both Coffee and Tea?
There are 27 students who drink both coffee and tea.
Explanation
80 students drink tea.68 students drink coffee.19 students drink neither coffee nor tea.
To find the number of students who drink both coffee and tea, we use the principle of inclusion-exclusion:
Total students = 140Number of students who drink tea (T) = 80Number of students who drink coffee (C) = 68
Using the principle of inclusion-exclusion: (T ∪ C) = T + C – (T ∩ C)
140 = 80 + 68 – (T ∩ C)
(T ∩ C) = 80 + 68 – 140 (T ∩ C) = 148 – 140 (T ∩ C) = 8
Now, to find the correct count, we adjust for the students who enjoy both coffee and tea:
(80 + 68) – 140 + 19 = 27
So, there are 27 students who enjoy both coffee and tea in this class.
So, 27 students enjoy both coffee and tea in this class.
What is Set Theory?
Set theory is a foundational branch of mathematics that deals with collections of objects, called sets, and the relationships between them. Developed in the late 19th and early 20th centuries by mathematicians such as Georg Cantor and Ernst Zermelo, set theory provides a formal framework for discussing and analyzing mathematical concepts across various disciplines.
Here are some key concepts and components of set theory:
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Sets: Sets are collections of distinct objects, called elements or members. These objects can be anything: numbers, letters, shapes, or even other sets. Sets are typically denoted by curly braces {} and their elements are listed within, separated by commas. For example, {1, 2, 3} is a set containing the numbers 1, 2, and 3.
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Elements: Objects contained within a set are called elements. An element either belongs to a set or does not belong to it.
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Subset: A set A is a subset of another set B if every element of A is also an element of B. This relationship is denoted as A ⊆ B. The empty set (∅) is a subset of every set.
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Union and Intersection: The union of two sets A and B, denoted by A ∪ B, is the set of all elements that are in A, in B, or in both. The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are in both A and B.
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Complement: The complement of a set A, denoted by A’, is the set of all elements not in A but in the universal set containing A.
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Cartesian Product: Given two sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a is in A and b is in B.
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Cardinality: The cardinality of a set is the number of elements it contains. It can be finite or infinite.
Set theory provides the foundation for various branches of mathematics, including algebra, calculus, and logic. It is widely used in mathematical logic, computer science, and many other areas of mathematics and applied sciences.
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Categories: Math
