91Ó°ÊÓ

Consider the function \(f:\\{1,2,3,4\\} \rightarrow\\{1,2,3,4\\}\) given by $$f(n)=\left(\begin{array}{llll}1 & 2 & 3 & 4 \\\4 & 1 & 3 & 4\end{array}\right)$$ (a) Find \(f(1)\). (b) Find an element \(n\) in the domain such that \(f(n)=1\). (c) Find an element \(n\) of the domain such that \(f(n)=n\). (d) Find an element of the codomain that is not in the range.

For each sentence below, decide whether it is an atomic statement, a molecular statement, or not a statement at all. (a) Customers must wear shoes. (b) The customers wore shoes. (c) The customers wore shoes and they wore socks.

Suppose \(P\) and \(Q\) are the statements: \(P\) : Jack passed math. Q: Jill passed math. (a) Translate "Jack and Jill both passed math" into symbols. (b) Translate "If Jack passed math, then Jill did not" into symbols. (c) Translate \({ }^{\prime \prime} P \vee Q^{\prime \prime}\) into English. (d) Translate \({ }^{\prime \prime} \neg(P \wedge Q) \rightarrow Q^{\prime \prime}\) into English. (e) Suppose you know that if Jack passed math, then so did Jill. What can you conclude if you know that: i. Jill passed math? ii. Jill did not pass math?

The following functions all have domain \\{1,2,3,4,5\\} and codomain \(\\{1,2,3\\} .\) For each, determine whether it is (only) injective, (only) surjective, bijective, or neither injective nor surjective. (a) \(f=\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 1 & 2 & 1\end{array}\right)\). (b) \(f=\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 3 & 1 & 2\end{array}\right)\). (c) \(f(x)=\left\\{\begin{array}{ll}x & \text { if } x \leq 3 \\ x-3 & \text { if } x>3\end{array}\right.\)

The following functions all have domain \\{1,2,3,4\\} and codomain \(\\{1,2,3,4,5\\} .\) For each, determine whether it is (only) injective, (only) surjective, bijective, or neither injective nor surjective. (a) \(f=\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 1 & 2 & 5 & 4\end{array}\right)\). (b) \(f=\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 2\end{array}\right)\). (c) \(f(x)\) gives the number of letters in the English word for the number \(x .\) For example, \(f(1)=3\) since "one" contains three letters.

Find a set of largest possible size that is a subset of both \\{1,2,3,4,5\\} and \\{2,4,6,8,10\\} .

Determine whether each molecular statement below is true or false, or whether it is impossible to determine. Assume you do not know what my favorite number is (but you do know that 13 is prime). (a) If 13 is prime, then 13 is my favorite number. (b) If 13 is my favorite number, then 13 is prime. (c) If 13 is not prime, then 13 is my favorite number. (d) 13 is my favorite number or 13 is prime. (e) 13 is my favorite number and 13 is prime. (f) 7 is my favorite number and 13 is not prime. (g) 13 is my favorite number or 13 is not my favorite number.

In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statements? (a) The square and the triangle are both blue. (b) The square and the triangle are both green. (c) If the triangle is not green, then the square is not blue. (d) If the triangle is green, then the square is blue. (e) The square is not blue or the triangle is green.

Find a set of smallest possible size that has both \\{1,2,3,4,5\\} and \\{2,4,6,8,10\\} as subsets.

Write out all functions \(f:\\{1,2\\} \rightarrow\\{a, b, c\\}\) (in two-line notation). How many functions are there? How many are injective? How many are surjective? How many are bijective?