Question

Relations and Functions

Usual symbols for the above are;

• Relations: R₁, R₂, S, T, etc
• Functions: f, g, h, etc. But remember a function is a special kind of relation so it might turn out that a Relation, R, is a function, too.

Relations

To understand the symbolism better, let’s say the domain of a relation, R, is A = { a, b , c} and the Codomain is B = { 1,2,3,4}.

Here is the relation: a R 1,     b R 4,     b R 3, and    c R 2.

We could also write (a,1) ε R, (b,4) ε R, (b, 3) ε R and (c,2) ε R.

Writing R as a set of ordered pairs, we have, R = { (a,1), (b,4,), (b,3), (c,2)}

Functions:

I will use a numerical function to illustrate symbolism used here, but remember a function, f, need not have any numbers in it.

Let’s say we have a function f that takes a number in its domain multiplies it by 2 and then subtracts 5.   We might write f(x) = 2x – 5.   Let’s also say that the domain of this function is A = { 1,2,3,4,5} and the Codomain is B = {-5, -4, -3, -2, -1, 0, 1,2,3,4,5,6}.   The Range is not the entire codomain.

Some ways of expression this function are the following:

1 f -3,   2 f -1, 3 f 1 and so on. This way is not the usual way, however.

The following are more standard ways:

f = {(1, -3), (2, -1), (3, 1), (4, 3), (5, 5) or we might write

(1, -3) ε f,    (2, -1) ε f,     (3,1) ε f,     (4,3) ε f, and (5,5) ε f.

Or in a more familiar way you might write: f(1) = -3, f(2) = -1, f(3) = 1, f(4) = 3 and f(5) = 5.

You complete this table for 10 extra credit points

Send only these last three pages to me via email by the announced due date . The name of the file should be YourLastName_ section 4.1

	State the Domain and Codomain.	Express relationship for 4 ordered pairs	Express the relationship using notation like a R b	Is the relationship a function? If the answer is yes, explain why. If the answer is No, give me an example of a domain element going to more than one Codomain element.	Give 4 ordered pairs in the inverse relationship. Two should be in the ordered pair notation and two in the R-1 notation.
This first row is just an example. The domain is the set of letters in the alphabet and the Codomain is the set of colors.   The relation R means that a letter in the alphabet is the first letter in a color name.	For the one to the left the domain is D = {all letters in alphabet} and the Codomain is G = {all colors}.	(r,red) ε R, ( b, blue) ∈ R, (p, pink) ∈ R and (y, yellow) ∈ R	r R red, p R pink g R green o R orange	No, R is not a function since (b, black) and (b, blue) both ∈ R	(pink,p) ∈ R-1 (yellow, y) ∈ R-1 Green R-1 g Orange R-1 o
A number 1 – 26 is associated with the letter in the alphabet in that location. Use S for the relationship symbol here.
A person is associated with his/her birth year. Use B as the relationship symbol
	State the Domain and Codomain.	Express relationship for 4 ordered pairs	Express the relationship using notation like a R b	Is the relationship a function? If the answer is yes, explain why. If the answer is No, give me an example of a domain element going to more than one Codomain element.	Give 4 ordered pairs in the inverse relationship. Two should be in the ordered pair notation and two in the R-1 notation.
A person is associated with his/her phone number. Use P for the relationship symbol
A number is associated with its cube. Use C for the relationship symbol
A number from N is associated with 0 if it is even and 1 if it is odd. Call the relationship R.
A number from N is associated with its remainder when that number is divided by 3. Use M for the relationship symbol.
	State the Domain and Codomain.	Express relationship for 4 ordered pairs	Express the relationship using notation like a R b	Is the relationship a function? If the answer is yes, explain why. If the answer is No, give me an example of a domain element going to more than one Codomain element.	Give 4 ordered pairs in the inverse relationship. Two should be in the ordered pair notation and two in the R-1 notation.
A real number is associated with its reciprocal. Use L for the relationship symbol.
A state in the US is associated with the year in which it was admitted to the Union. Use A for the symbol.
A president of the US is associated with his/her order of presidency. Use P for the relationship. ASI type this file it is Nov 7 so I do not know if we will have our first female president or not.

Answer 1