Question

1. Let A={1,2,3,4} and B={a,b,c}.
(i) Write down a function from A to B
(ii) Write down a different function from A to B
(iii) Write down a relation from A to B which is not a function
(iv) How many functions are there from A to B?

2. Determine whether each of the statements that follow are true or false. If the statement is true, give a reason. If it is false, give a counterexample (i.e. a function for which the statement is false)
(i) The domain of every function is a subset of R.
(ii) If f:R=>R is a function and xER then f(2x) = 2f(x)
(iii) If h:A=>B is a function and a, bEA then h(a) = h(b) implies that a=b

Answer 1

Answer on Question #47438 – Math – Calculus

1. Let $A = \{1,2,3,4\}$ and $B = \{a,b,c\}$.

(i) Write down a function from A to B

(ii) Write down a different function from A to B

(iii) Write down a relation from A to B which is not a function

(iv) How many functions are there from A to B?

2. Determine whether each of the statements that follow are true or false. If the statement is true, give a reason. If it is false, give a counterexample (i.e. a function for which the statement is false)

(i) The domain of every function is a subset of R.

(ii) If $f: R \Rightarrow R$ is a function and $x \in R$ then $f(2x) = 2f(x)$

(iii) If $h: A \Rightarrow B$ is a function and $a, b \in A$ then $h(a) = h(b)$ implies that $a = b$

Solution.

1. A relation is a set of ordered pairs $(x, y)$.

A function is a relation that does not contain two pairs with the same first component.

(i) $\{(1, a), (2, b), (3, c), (4, a)\}$

(ii) $\{(1, a), (2, a), (3, a), (4, a)\}$

(iii) $\{(1, a), (1, b), (2, b), (3, c), (4, a)\}$

(iv) For each element from $A$ we can choose each element from $B$. If function $f: A \to B$, $A = \{a_1, a_2, a_3, a_4\}$, then we count the number of possible functions from $A$ to $B$, $a_1 \to |B|$, $a_2 \to |B|$, $a_3 \to |B|$, $a_4 \to |B|$, i.e., this number is $|B| \cdot |B| \cdot |B| \cdot |B| = |B|^{|A|} = 3^4 = 81$.

Thus, there are 81 different functions from $A$ to $B$.

2.

(i) False. Set A can be a set of arbitrary elements, for ex. "red", "blue", "black".

(ii) False. For ex. $f(x) = x^{2} \rightarrow f(2x) = 4f(x) \neq 2f(x)$.

(iii) False. For ex. $f(x) = \sin(x)$. $f(a + 2\pi) = f(a)$ but $a \neq a + 2\pi$.

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