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,

large crystals result from very slow cooling.

evaporation of water: this would produce an evaporite which is a sedimentary rock. like a salt pan in a desert. not possible to have volcanic rocks originate this way.

crystalization in molten earth: slow cooling over time. (hundreds to millions of years)

crystalization at earth's surface: this results in fast cooling. (a few hours to a few months or years)

crystalization from cooling off hot water: that would be a hydrothermal deposit. while associated with volcanics they are not volcanic rocks themselves and usually occur fairly rapidly.