Logic.pdf

Logic:
> is the science and art of correct
thinking.
Aristotle: is generally regarded as the
father of Logic. ( 382 – 322 BC)
PROPOSITION: or (Statement)
- Is a declarative sentence which is either
true or false, but not both.

Operations on Propositions
• The main logical connectives
such as conjunctions,
disjunctions, negation,
conditional and biconditional.

• George Boole in 1849 was appointed
as chairperson of mathematics at
Queens College in Cork, Ireland.
Boole used symbols p, q and
symbols ʌ , v , ~ , , ,
to represent connectives.

• Logic connectives and symbols:
• Statement connective symbolic form Type of statement
• Not p not ~ p negation
• p and q and p ʌ q conjunction
• p or q or p V q disjunction
• If p , then q if … then p q conditional
• P if and only if q if and only if p q biconditional

TRUTH VALUE & TRUTH TABLES
• Truth Value of a simple statement is either
true ( T) or false (F).
• Truth Table is a table that shows the truth
value of a compound statement for all
possible truth values of its simple statement.

• Conjunction. The conjunction of the proposition
p and q is the compound proposition.
Symbolically p ^ q.
Truth table
p q p ^ q
T T T
T F F
F T F
F F F

EXAMPLES:
1. 2 + 3 = 5 and 3 x 2 = 6.
p = t q = t
2. 6 x 7 = 42 and 3 x 6 = 19.
p = t q = F
3. 2 x5 = 11 and 8 x 8 = 64.
p = F q = t
4. 4 x 5 = 21 and 5 x 5 = 26.
p = F q = F

• Disjunction. p or q symbolically p v q.
truth table
p q p v q
T T T
T F T
F T T
F F F

Examples:
1. 3 or 5 are prime numbers.
p = t q = t
2. 2 x 5 = 10 or 2 x 6 = 13.
p = t q = F
3. 3 x5 = 16 or 4 x 6 = 24.
p = F q = T
4. 20 x 2 = 30 or 10 x 4 = 50.
p = F q = F

• Negation. The negation of the proposition p is
denoted by ~ p.
• Truth table
p ~ p
T F
F T

• Examples:
Peter is a boy. ( p) true
• Peter is not a boy. ( ~ p ) false
Dog is a cat. ( p) false
Dog is not a cat. ( ~ p) true

• Conditional. The conditional of the proposition p
and q is the compound proposition if p then q.
p q .
truth table
p q p q
T T T
T F F
F T T
F F T

Examples:
1. IF VENIGAR IS SOUR , THEN SUGAR IS SWEET.
p = t q = t
2. IF 2 + 5 = 7 , THEN 5 + 6 = 12
p = t q = F
3. IF 14 – 8 = 10, THEN 20 / 2 = 10
p = F q = F
4. IF 2 X 5 = 8 , THEN 40 / 4 = 20.
p = F q = F

BICONDTIONAL . The biconditional of the
proposition p and q is the compound
proposition p if and only if q. Symbolically
p q.
truth table
p q p q
T T T
T F F
F T F
F F T

Examples:
1. 3 is an odd number if and only if 4 is an even number.
p = t q = t
2. 10 is divisible by 2 if and only if 15 is divisible by 2.
p = t q = F
3. 8 is a prime number if and only if 2 x 4 = 8.
p = F q = T
1. 8 – 2 = 5 if and only if 4 + 2 = 7.
p = F q = F

LOGIC PUZZLE
A logical puzzle is a problem that can be solved
through deductive reasoning.

1. Draw the missing tile from the below matrix.

answer:
Sol.
Each lines contains a heart, spade and club
with one of the symbols inverted.

2. Feed me and I live, yet give me a drink and I die.

ANSWER: 19
Explanation : As you move diagonally down,
numbers follow the sequence of Prime Numbers.

4. What number should replace the question mark?

•Answer
Ans: 17.Sol.
It is the sum of the two digits(9 + 8) in the quadrant
opposite.

Answer
:
9
: The number at the center of each triangle equals the sum of the
lower two numbers minus the top number

5. A large volume of water is gushing through a pipe
which narrows at the outlet. At which point, A, B, C or
D will the water flow fastest?

Ans: C.
Sol.
The water flows fastest at the narrowest point.

4. How many lines appear below?

5. What number should replace the question
mark?

Ans: 0.
Sol.
Looking at lines of numbers from the top : 9×8
= 72; 72×8 = 576; 576×8 = 4608;
•Answer

6. Which three of the four pieces below can be fitted
together to form a perfect square?

7. Which does not belong in this sequence?

Ans: E.
Sol.
The black dot is moving one corner clockwise at
each stage and the white dot is moving one corner
anti-clockwise at each stage.

8. Complete two eight-letter words, one in each circle,
and both reading clockwise. The words are synonyms. You
must find the starting points and provide the missing letters.

Ans: ESCALATE, HEIGHTEN.
Sol.
ESCALATE, HEIGHTEN.

9.How many hexagon (six sided figures) can you find here?

Ans: 21.
Sol.
There are 15 small hexagons and 6 large
ones. The last shape in the bottom row is a
pentagon.

10.Find the relationship between the numbers in the top row
and the numbers in the bottom row, and then determine
the missing number.

Ans: The missing number is 12.
Sol.
When you look at the relationship of each set of
top and bottom numbers, you will see that
12 is 2 x 6,
24 is 3 x 8,
36 is 4 x 9,
45 is 5 x 9,
60 is 6 x 10, and
84 is 7 x 12.

Ans: E.
Sol.
All the others have four twigs on the left side
branch and three twigs on the right side.
Option E is the other way round.

Answer: 49.
Sol.
( 73 + 25) ÷ 2 = 49.

Ans: 10.
Solution:
Each diagonal line of numbers, starting with the top
left hand corner number, increases by 1 each time,
ie :

Kenken Puzzles:
- Is an arithmetic – based logic puzzle that was invented by the Japanese
mathematics teacher Tetsuya Miyamoto in 2004. the noun “ ken” has
“knowledge” and “awareness” as synonyms.
- Rules for solving a kenken puzzle:
- For a 3x3 puzzle, fill in each box ( square) of the grid with one of the
numbers 1,2,3.
- For a 4x4 puzzle, fill in each box ( square) of the grid with one of the
numbers 1,2,3,4.

2 1 3
3 1
1 3 2
b a c
c a
a c b

2 1 3 4
3 2 4 1
1 4 2 3
4 3 1 2
D

Magic Square
How To Find A Magic Square Solution?
Solving a 3 by 3 Magic Square
We will first look at solving a 3 by 3 magic square puzzle. First off, keep in mind
that a 3 by 3 square has 3 rows, and 3 columns.
When you start your 3 by 3 square with you will either choose or be given a set
of nine consecutive numbers start with to fill the nine spaces.
Here are the steps:
•List the numbers in order from least to greatest on a sheet of paper.
•Add all nine of the numbers on your list up to get the total. For example, if the
numbers you are using are 1, 2, 3, 4, 5, 6, 7, 8. and 9, you would do the
following: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. So 45 is the total.
•Divide the total from Step 2 by 3. 45 divided by 3 = 15. This number is
the magic number. (The number all rows, columns, and diagonals will add up
to.)
•Go back to your list of numbers and the number in the very middle of that list
will be placed in the center of the magic square.

•x represents the number in the center square so for our example, x is equal
to 5.
•Place the number x + 3 in the upper right-hand square.
•Place the number x + 1 in the upper left-hand square.
•Place the number x - 3 in the lower right-hand square.
•Place the number x -1 in the lower upper left-hand square.
•For the last step, fill in the remaining squares keeping in mind the magic
number is 15. So that means all the rows, columns, and diagonals need to
add up to 15. You should have no problem figuring where to place the
remaining numbers keeping this in mind. The complete magic square is in
picture below.

A Magic Square 4 x 4 can be considered as the King of all the
Magic Squares, for its an array of 16 numbers which can be added
in 84 ways to get the same Magic Sum.
Example 1
As usual we are going to make a magic square with the first
natural numbers, the magic sum of first 16 natural numbers is 34.
Going by our formula.
Natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

STEP– 1
Draw 4 x 4 square grid and put small ‘dots’ in the boxes along both diagonals. Start
counting from the topmost left and fill the un dotted boxes with the respective
terms. Although the dotted boxes are counted the term corresponding to that box is
not written there and the adjacent box is filled in the order.( Here the crossed cells
are not filled even though they are counted with numbers written in it,)
STEP– 2
Now start filling from the lower most right cell, the boxes are counted using the
same NN towards left. The dotted boxes are then filled with the numbers
corresponding to that box.
Alternate Method. The same result can be obtained if we fill all the cells in the order
with out putting any vacant cells along the diagonals and then reversing the filling
in these diagonals.
Here the Magic Sum
a) Along Rows = 34 b) Along Columns = 34
c) Along Diagonals = 34

5x5 Magic Squares
This is the basic 5x5 magic square.
It uses all the numbers 1-25 and it
adds up to 65 in 13 different ways:
•All 5 vertical lines add up to 65
•The two diagonals both add up to 65
•Finally you can add up the four
corners and the number in the middle
to get 65.
•All 5 horizontal lines add up to 65

1 8
5 7
4 6
3
2
How to lay out a 5x5 Magic Square
Have another look at the way the numbers are set out in the original square. It uses all the
numbers 1-25, and if you follow the numbers round in order you'll see they appear in this pattern:
1,2,3,4 and 5 are in a diagonal line, which goes off the top and comes back at the bottom, then
goes off the right and comes back on the left. Once the first five numbers are in place, there's no
empty place to put number 6.
The rule is to put the 6 UNDER the 5, and then continue putting numbers in another diagonal
line:

Again you'll see that the numbers 6-10 are in a diagonal which
goes round until there's no space for the 11. So the 11 goes under
the last number which was 10. If you keep going, you'll fill the
whole grid with the numbers 1-25 and make the basic 5x5 magic
square.
1 8
5 7
4 6
3
2 9

1 8 15
5 7 14 16
4 6 13
10 12 3
11 2 9

17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9

STEP 4
8 1 6
3 5 7
4 9 2
17 10 15
12 14 16
13 18 11
STEP 2
STEP 3
6 X 6 MAGIC SQUARE
26 19 24
21 23 25
22 27 20
35 28
STEP 4
33
30 32 34
31 36 29
STEP 1

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