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Which of the following are propositions? Of those which aren’t, explain why.

The baby is laughing.

What’s that noise?

1 + 1 = 3

Get out of here!

He ran in the race but slipped on a banana peel.

What a glorious day it is!

Santa Claus is jolly.

To fail to achieve the impossible is not to fail.

Please pay attention.

All triangles have four sides.

The next Prime Minister will be a woman.

What are you thinking?

If it rains then there is moisture in the air.

I wish to be immortal.

Neither circumstances nor criticism will prevent my progress.

Won’t you close the door?

Santa Claus is a fictional character.

Let x mark the spot.

Your wish is my command.

Define ‘proposition’ to mean ‘bearer of one truth’.

] Carry out the following calculations and conversions by hand (showing all your working). (a) Compute the sum (1011010)2 + (11011)2 (without changing the base). (b) Compute the sum (135)8 + (2357)8 (without changing the base). (c) Convert (197)10 to octal. (d) Convert (20A5.76)12 to decimal.


Find, showing all working, a formula for the n-th term tn of the sequence (tn) defined by t1 = 5; tn = −7tn−1/3, n ≥ 2.


state the value of x after the statement if P(x) then x = 1 is executed , when P(x) is the statement =x>1 , " if the value of x when this statement is reached is
~(P⠆ ’q) ⠆ ’ p
simplify by laws

Give the power set of the following sets.

(a) /0 (b) {1} (c) {1,2} (d) {1,2,3}



. Let p and q be the propositions p: He is rich q: He is happy Write the following propositions using p and q and logical connectives.

a. If he is rich, then he is unhappy.

b. He is neither rich nor happy.

c. It is necessary to be poor, in order to be happy.

d. He is either rich or happy (or both).


Show following equivalence without considering the truth table.

(𝑝̅ ∧( 𝑞̅∧𝑟)) ∨(𝑞 ∧𝑟) ∨(𝑝 ∧𝑟)↔𝑟


Prove by contradiction that for any integer n if n2 is odd then n is odd.


Write the converse, inverse, and contrapositive of the statement “If

5 is an odd number, then it is a prime number.”


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