UGC NET Paper 1 Mathematical Reasoning & Aptitude.pdf
Pptmath
1. SWBAT analyze, create and
identify sets, subsets and ways
to express them.
Standards: A.A.29 and A.A.30
POD: Ryan Gosling tells Michael that -4 is an integer and a
whole number. Is he right or wrong? Explain why, and if not,
state the all the number groups -4 belongs to.
???????
????
2. Solution
Ryan Gosling is WRONG, because -4 is an
integer, but it is not whole. Whole
numbers are {0,1,2,3,4…}, and do not
include negative numbers. -4 belongs to
rational and integer numbers.
5. Obviously, there are more Canadians
than Ontarians in the world, and not
all Canadians are from Ontario
6. But, every single person from
Ontario is from Canada
ALL CANADIANS
People from Ontario
7. So what does this mean? (Other
than your new knowledge that Ryan
Gosling is from Ontario)
A set is a collection of elements or
members. A set is denoted by {}.
For example: {3,4} {2} {7,9,12,16}
So, a set of all Canadian territories and
provinces is {Manitoba, Alberta, Ontario,
Saskatchewan, Quebec, Prince Edward’s
Island, New Brunswick, Nova Scotia,
Newfoundland and Labrador}
8. A subset consists of elements from any given
set. If A={1,2,3}, and B={1,3}, then B is a subset
of A. A subset is indicated by ⊆ In this case,
A⊆ B. If you want to indicate that A is not a
subset of B, you would indicate that with ⊄.
For example:
A={4,5} B={4}
B⊂A, because B only has some members, but not all the
members in A.
A subset of the Canadian territories can be {Ontario}, only,
or {Ontario, Quebec}. “Ryan Gosling is from Canada, meaning
{Manitoba, Alberta, Ontario, Saskatchewan, Quebec, Prince
Edward’s Island, New Brunswick, Nova Scotia, Newfoundland and
Labrador}, but more specifically, {Ontario}.
However, a subset can have as many members
which are in the original set, even 0! Subsets
with no members is represented by {} or Ø are
known as null sets.
9. A power set is the set of all subsets
of a set.
You can calculate the number of
subsets in a set by using the
extremely simple formula 2^n, where
n= the amount of terms in a set.
For example:
How many subsets are in {13, 25, E, W, F, 9, 4}?
There are 7 terms in the set, so you could plug in 7 for
n, making 2^7. 2^7 is equal to 128 subsets.
11. However, sometimes you may be asked to
list only the PROPER SUBSETS. What
are proper subsets? Proper subsets are
subsets that do NOT include all
elements in a set. Therefore, the
number of proper subsets in {13, 25, E,
W, F, 9, 4} is 127 subsets, excluding
the improper subset of {13, 25, E, W, F,
9, 4}.
12. If a set has infinite members, it is
called an infinite set. If it has a
certain amount of members, it is a
finite set.
Examples of infinite sets include
{2,4,6,8,10,12…}. Infinite sets are noted
by the “…”, known as ellipses. However,
ellipses can be used for finite sets when
you don’t feel like writing the complete
set, such as {a,b,c,d…w,x,y,z}.
14. Done yet? No? It’s infinite you say?
This is a universal set, or universe is the
largest set possible with the subject you
are using. It doesn’t necessarily have to
be infinite like numbers though.
For example, a universal set could be {months of the
year} which would stand for {January, February, March,
April, May, June, July, August, September, October,
November, December}. Months aren’t infinite but 12
months is the most you can have, so it’s universal.
15. If you had a subset of the universal set,
being P = {January, May, July, August,
October, December}, them the
complement of the set would be P’ (P
prime) = {February, March, April, June,
September, November}. The complement
of the set contained elements of the
universal sets of {months of the year} not
already given in a subset.
Please note that it’s
spelled complement,
not compliment. If
we wanted a
compliment, just
comment on our
great looks
A complement of a set is everything that
is in the universal set but not inside the
subset. In this example it’s P.
16. LETS TAKE A LOOK AT A REAL
WORLD EXAMPLE!
In this
diagram, the
universal set is
Canadians while the
subset is Ontarians.
Complement of
Ontarians is
Canadians that are
not Ontarians
which is {Alberta,
British Columbia,
Saskatchewan,
Manitoba…Quebec}
ALL CANADIANS
People from Ontario
17. Try it!
Ryan Gosling has to choose what color to dye his hair. His preferred
choice is any color of the rainbow. When Ryan goes to the salon, his hair
stylist tells him that they don’t carry indigo and violet. What colors can
Ryan Gosling dye his hair? Create a universal set, a subset of the universal
set, and find the complement of the set. Don’t forget to use variables!
How would you like
it done sir?
18. Solution
Answer:
Let C = {colors of the rainbow} and let N
={indigo, purple}
Since Ryan cannot dye his hair indigo or
purple, the answer is the complement of N.
You would express the complement of N
using N’.
N’ = {red, yellow, blue, green, orange}
Ryan Gosling can dye his hair red, yellow,
blue, green, or orange.
19. How else could you express sets?
The form {} In expressing sets is known
as roster form. You could also express
a set with set-builder notation.Set
builder notation is a way of “building a
set” by listing the requirements and
properties of the elements of the set.
Example:
20. Another way to express sets is through
interval notation. An interval is a
connected subset of numbers. Interval
notation is an alternative to expressing your
answer as an inequality.
In interval notation, “(“, known as a soft
bracket indicates "not included" or "open".
“[“, known as a hard bracket includes means
"included" or "closed".
Example: (a, b) is interpreted as a < x <
b where the endpoints are NOT included.
[a, b] is interpreted
as a < x < b where the endpoints are
included.
21. Try it again!
Ryan Gosling is in the running for the sexiest
man alive award. His competition, however,
is Joseph Gordon-Levitt and Benedict
Cumberbatch. A panel of 200 people will
vote on the winner. At the moment, Joseph
Gordon-Levitt has 64 votes, Benedict
Cumberbatch has 63 votes while Ryan
Gosling has 66 votes. In set builder
notation, what number of votes from the
remaining votes does Ryan Gosling need to
become the sexiest man alive?
22. Solution
Answer: P = {x|x is a whole number and x ≥ 3}
Out of 200 panelists, 64 of them voted for
Joseph Gordon-Levitt, 63 voted for Benedict
Cumberbatch, and 66 voted for Ryan Gosling.
So, in total, 193 voted. If all 7 remaining
panelists all voted for Joseph Gordon-Levitt or
they all voted for Benedict Cumberbatch, then
Ryan Gosling would lose. If Ryan Gosling gets at
least 3 votes, even if the other 4 voted for
Joseph Gordon-Levitt, Ryan Gosling would still
win by 1 vote.
I hope they
are not that
upset that
they lost
23. Group Work
Ryan Gosling is constructing a swimming pool in his backyard. He has already decided that it
will be 5 feet deep. He wants the length of the pool to be at least 1008 inches but no more than
1596 inches, and the width to be less than 150 yards but larger than 75 yards. His agent tells him
that it will cost $10 per cubic foot of pool he wants made and only whole numbered distances will
be accepted. There will be a 15% tip required on top of the final price.
1.
2.
3.
4.
5.
6.
7.
In interval notation, describe what lengths of the pool Ryan Gosling would accept. Make a
number line to show the data. Let L = the length in feet.
In interval notation, describe what widths of the pool Ryan Gosling would accept. Make a number
line to show the data. Let W= the width in feet.
What is the greatest number of cubic feet that could be in the pool?
What is the least number of cubic feet that could be in the pool?
What is the greatest amount (tip included) that Ryan Gosling could pay for creating the pool?
What is the least amount (tip included) that Ryan Gosling could pay for creating the pool?
If Ryan Gosling decides he wants a square shaped pool that is the largest size possible, what price
will he pay for it?
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