Problem 32 - Discrete Mathematics with Applications

Question

Mark each sentence as true or false. Assume the composites and inverses are defined: Every bijection is invertible.

Step-by-Step Solution

Answer

The statement "Every bijectible (bijective) function is bijective" is trivially true, as a bijective function, by definition, is a function that is both one-to-one and onto.

Step 1: Statement 1

Every bijectible function is bijective. Since the term "bijectible" is not standard, we will consider it to be a typo and assume that the correct term is "bijective". Bijective functions by definition are functions that are both injective (one-to-one) and surjective (onto).

Step 2: Analysis of Statement 1

Given that the statement is: Every bijective function is bijective, we can see that the statement is asking if a function with the properties of being bijective (one-to-one and onto) is indeed bijective. This is trivially true as a bijective function, by definition, is a function that is both one-to-one and onto.

Step 3: Conclusion

Based on the analysis, the statement "Every bijectible (bijective) function is bijective" is true.

Key Concepts

Bijective functions

Bijective functions are a crucial concept in Discrete Mathematics, enabling the manageable mapping between two sets. A function is termed "bijective" when it satisfies two primary properties: injective and surjective. This means each element from the domain maps to a unique element in the codomain, and every element of the codomain is the result of the mapping from the domain.

Mathematically speaking, if there exists a function, say \( f : A \rightarrow B \), it is bijective only if:

• Injective: For any two elements \( a_1, a_2 \in A \), if \( f(a_1) = f(a_2) \), then \( a_1 = a_2 \).


• Surjective: For every \( b \in B \), there exists an \( a \in A \) such that \( f(a) = b \).

With these attributes, a bijective function ensures that an inverse function exists, which enables one to navigate backwards through the mapping from the codomain to the domain.

Injective and surjective functions

Two pivotal terms related to functions are injective and surjective. These properties help define the relationship between two sets in the context of functions. A function is injective, or one-to-one, when every element of the domain maps to a distinct element in the codomain. This implies that no two different elements from the domain can point to the same element in the codomain. Essentially, different inputs lead to different outputs.

On the flip side, a function is surjective if each element in the codomain is covered by the mapping from at least one element in the domain. This means the function's reach covers the entire codomain, ensuring nothing is left out.

To recollect, injectivity ensures uniqueness of mappings, while surjectivity ensures completeness. Together, these properties are foundational in understanding the structure and behavior of functions.

Mathematical logic

Mathematical logic forms the backbone of reasoning in mathematics, providing the structure and language for constructing proofs and solving equations. It involves applying strict logical reasoning, ensuring that every step in the process maintains correctness by following the established rules.

Using logical connectors like "if", "then", "and", and "or" allows mathematicians to structure arguments and validate ideas. For instance, the statement "Every bijective function is bijective" would check out trivially true due to the definition of bijection itself. Mathematical logic helps clarify and verify such concepts, leading to sound conclusions.

Conclusively, by grounding mathematical theory in logical reasoning, mathematical logic ensures that conclusions drawn are not only accurate but also compelling and universally accepted.