For each of the following relations on the set {0, 1, 2, 3}. • r1 = {(0, , , , 3)} • r2 = {(1, , , 3)} • r3 = {(0, , , , , , 3)} • r4 = {(0, , , , , , 3)} (a) which of these relations are reflexive? (b) which of these are symmetric? (c) which of these are anti-symmetric? (d) which of these are transitive?

(a)  

(b)

(c)

(d)

Step-by-step explanation:

Since,

A relation R defined on A is called reflexive if,

∀ x ∈ A, (x, x)∈ R  

It is called symmetric,

∀ x, y ∈ A, if (x, y)∈ R then (x, y)∈ R such that x ≠ y,

It is called anti symmetric,  

∀ x, y ∈ A, if (x, y)∈ R then (x, y)∈ R such that x = y,

It is called transitive,

∀ x, y, z ∈ A, if (x, y)∈ R and (y, z)∈ R then (x, z)∈ R such that x ≠ y,

Given set,

{0, 1, 2, 3},

Also, the relation defined on the set are,

• = {(0, 0),(1, 1),(2, 2),(3, 3)}

• = {(1, 1),(2, 2),(3, 3)}

• = {(0, 0),(1, 1),(2, 0),(2, 2),(2, 3),(3, 3)}

• = {(0, 0),(0, 1),(1, 0),(1, 1),(2, 2),(3, 3)}

Hence, by the above explanation it is clear that,

Relations which are reflexive,

Relations which are symmetric,

Relations which are anti-symmetric,

Relations which are transitive,

3

step-by-step explanation:

in standard form:

a= dictionaries

b= maps

you have to buy 3 dictionaries, so put a 3 next to the a (dictionaries) so that you can multiply them by each other.

a3+b= 60 (the amount of money you have)

we already know the cost of a=15 so replace a with $15

($15 x 3)+b=60

multiply 15 and 3 to get $45 and replace it with $45

45+b=60

and then replace b with 15 to get the total:

45+15=60

that means you have $15 left after buying 3 dictionaries.

15 divided by the cost of a map ($5) equals 3.

so you can buy 3 maps.

sorry if it was confusing!

p ÷ r = tan(x)

step-by-step explanation:

given that there is a figure below shows a right triangle: a right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units.

graph is missing so i will create the graph as per given information.

the angle between the legs having lengths p units and q units is y degrees. the angle between the legs having lengths r units and q units is x degrees.

now we need to find about what is p ÷ r equal to.

so we can use trigonometric ratio tan(x)=[altitude]/[base] to find the value of p ÷ r, because p is the altitude and r is the base.

p ÷ r=tan(x)

hence final answer is p ÷ r=tan(x).