Q.8 The total number of mappings from the set {1,2} to the set {3, 4,5,6,7} is _____.
The total number of mappings from the set {1,2} to {3,4,5,6,7} is 25. This calculation arises from the fundamental principle of functions between finite sets, where each element in the domain maps independently to any element in the codomain.
Understanding Mappings
Mappings, or functions, assign each element from the domain set to exactly one element in the codomain set. For domain |A| = 2 elements and codomain |B| = 5 elements, the total count follows ∣B∣∣A∣=52=25. Element 1 has 5 choices, and element 2 has 5 choices, yielding 25 possible functions
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Step-by-Step Calculation
Calculate as follows: first element (1) maps to any of 5 options in {3,4,5,6,7}. Second element (2) independently maps to any of the same 5. Total mappings equal 5×5=25.
Types of Functions Explained
Functions from {1,2} to {3,4,5,6,7} include various types:
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Total functions: All 25 mappings qualify as total since every domain element maps.
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Injective (one-one): 5 × 4 = 20, as images differ.
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Surjective (onto): 0, since codomain has 5 elements but domain only 2.
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Bijective: 0, impossible due to unequal sizes.
This breakdown clarifies why total mappings encompass all possibilities beyond restrictions.
Common Exam Options Analyzed
In multiple-choice contexts, options might include:
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10: Mistake assuming injections only (ignores repetitions).
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20: Counts injections (5P2), excludes many-one functions.
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25: Correct total mappings.
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32: Error like 2^5, confusing direction.
Selecting 25 accounts for all valid assignments.
