91Ó°ÊÓ

When asked what it means to say that set \(A\) has the same cardinality as set \(B\), a student replies, " \(A\) are \(B\) are one-to-one and onto." What should the student have replied? Why?

a. If 4 cards are selected from a standard 52 -card deck, must at least 2 be of the same suit? Why? b. If 5 cards are selected from a standard 52 -card deck, must at least 2 be of the same suit? Why?

a. If 13 cards are selected from a standard 52 -card deck, must at least 2 be of the same denomination? Why? b. If 20 cards are selected from a standard 52 -card deck, must at least 2 be of the same denomination? Why?

In a group of 700 people, must there be 2 who have the same first and last initials? Why

Indicate whether the statements in parts (a)-(d) are true or false. Justify your answers. a. If two elements in the domain of a function are equal, then their images in the co-domain are equal. b. If two elements in the co-domain of a function are equal, then their preimages in the domain are also equal. c. A function can have the same output for more than one input. d. A function can have the same input for more than one output.

Let \(\mathbf{O}\) be the set of all odd integers. Prove that \(\mathbf{O}\) has the same cardinality as \(2 \mathbf{Z}\), the set of all even integers.

a. Given any set of four integers, must there be two that have the same remainder when divided by 3 ? Why? b. Given any set of three integers, must there be two that have the same remainder when divided by 3 ? Why?

All but two of the following statements are correct ways to express the fact that a function \(f\) is onto. Find the two that are incorrect. a. \(f\) is onto \(\Leftrightarrow\) every element in its co-domain is the image of some element in its domain. b. \(f\) is onto \(\Leftrightarrow\) every element in its domain has a corresponding image in its co-domain. c. \(f\) is onto \(\Leftrightarrow \forall y \in Y, \exists x \in X\) such that \(f(x)=y\). d. \(f\) is onto \(\Leftrightarrow \forall x \in X, \exists y \in Y\) such that \(f(x)=y\). e. \(f\) is onto \(\Leftrightarrow\) the range of \(f\) is the same as the co-domain of \(f\).

a. Given any set of seven integers, must there be two that have the same remainder when divided by 6 ? Why? b. Given any set of seven integers, must there be two that have the same remainder when divided by \(8 ?\) Why?

a. How many functions are there from a set with three elements to a set with four elements? b. How many functions are there from a set with five elements to a set with two elements? c. How many functions are there from a set with \(m\) elements to a set with \(n\) elements, where \(m\) and \(n\) are positive integers?