Generosity - Compassion

To move from occasional acts of generosity in an otherwise consumerist lifestyle to a

lifestyle of generous love requires the formation of a number of habits

Compassion: Seeing and feeling things from the perspective of the other person and

possessing the desire to do something to help them.

Identify one or two actions you will take to cultivate that habit in major spheres of

life

- Being sensitive to others needs

Generosity - Presence

To move from occasional acts of generosity in an otherwise consumerist lifestyle to a

lifestyle of generous love requires the formation of a number of habits

Presence: making ourselves available and attentive to others

Identify one or two actions you will take to cultivate that habit in major spheres of

life

- Listen to what the person is saying. Ask God quietly what He wants you to do.

Maths 101 revision - Lecture 9

Math 101 – Lecture 8

Basic Arithmetic

The set of real numbers R has the following properties a+b = b+a  R for a, b  R  a+ 0 = 0 + a, the additive identity.

For each a ( 0) R, there exists b  R such that ab = 1 where 1 is the multiplicative identity a*1 = 1*a = a.

Order for any two real numbers a, b  R  only one of the following is true:

a > b or

a = b or

a < b

If a, b, c  R and a > c, c > b then a > b

            If a, b, c  R and a > b then a + c > b + c

If a, b, c  R and a > b, if c > 0 then ac > bc

From these properties, together with the deankind axiom all the familiar results for real numbers.

Eg Index Laws a^m a . a . a .... a, m  IN

                        {          m times  }

            Extend to a^-m = 1/(am) = 1(a . a . a .... a) =              1/a . 1/a . 1/a .... 1/a

                                                            {          m times  }        {          m times  }

            Extend to a^(p/q), p, q   Z

Eventually extend to a^-σ , σ   R

Absolute value – a ‘distance measure’ on R

|a| = {a, a  0

            -a, for a < 0

Theorem (properties of absolute values)

1.      |a|  0 for all a   R, note  |a| = 0 if any only if a = 0

2.      |ab| = |a| * |b|, for all a, b  R

Cases

(i)                 |ab| = ab = |a| |b|     ie ab > 0 for [(i) a, b > 0

(ii) a, b < 0

(ii)               |ab| = ab = (-a)(-b)

3.      |a|^2 = a^2

4.      |a+b|  |a| + |b|, a, b  R

Proof: If a+b<0 then |a + b| = -(a + b)

                                                = (-a)  b),

Now – a  |a|, for any a  R

If a + b > 0 then |a + b| = a + b

But a  |a| and b  |b|

 |a + b|  |a| + |b|

5.      |a – b|  |a| - |b|, for any a, b   R

-||-

Bounded Sets

Infinite sets : 1) Q

                  2) Prime numbers

                  3) Z

      4) {x   Q: -1  x  1}

4) is a subset of an interval of the real number line.

[a, b] = {x^(R): a  x  b} – a closed interval.

[a, b] = { x^(R): a  x  b }

Also have [a, b] and [a, b]

Bounded sets: - A set S  R is bounded above if thre is K  R, such that K  for all x    S

      S is said to be bounded below.

                  If there is k    R such that x   k for all x    S

                        S is bounded if it bounded above and below.

*The least upper bound (supremum) for a bounded set S  R is K  R such that K  x for some x    S and any  > 0

*The greatest lower bound (infinum) for is K  R such that [ k  x, x  S

                                                                                                      K +  > x for some x  S and any  > 0