91Ó°ÊÓ

Using the pigeonhole principle, prove that the cardinality of a finite set is unique.

Mark each sentence as true or false. Assume the composites and inverses are defined: The composition of two bijections is a bijection.

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{i=1}^{5} \sum_{j=1}^{5} \delta_{i j} $$

Find the day of the week in each case. 234 days from Monday

A square matrix \(A\) is symmetric if \(A^{\prime \mathrm{T}}=A .\) What can you say about the elements of a symmetric matrix \(A\) ?

Let \(A\) be a square matrix. Prove that \(\left(A^{\mathrm{T}}\right)^{\mathrm{T}}=A\).

Prove. A bijection exists between any two closed intervals \([a, b]\) and \([c, d],\) where \(a< b\) and \(c< d\) . (Hint: Find a suitable function that works.)

Prove. The set of odd positive integers is countably infinite.

Prove. The set of integers is countably infinite.

If \(g \circ f\) is injective, then \(f\) is injective.