Problem $1[18$ pts $(3$ each$)]$: English to Logic

Express each sentence using logical operations $,,$ and the propositional variables a$,$ b$,$ and c

defined below. The use of the word $$or$$ in the sentences below always means inclusive or$.$

a The alarm is ringing

b The battery is low

c The system is on

i The alarm is ringing and the system is on

ii The alarm is not ringing but the system is on

iii The system is not on and the alarm is not ringing

iv There is no way that the alarm is ringing

v The alarm is ringing or the battery is low

vi Despite the fact that the alarm is ringing, the battery is low

Problem $2[15$ pts $(3$ each$)]$: Arithmetic & Logic

Evaluate each of the following statements as True or False. Show intermediate steps for full credit.

Try by simplifying only one operator $(,,>,=,...)$ at a time if the whole expression is difficult

to understand.

i $(7>=7)(8<5)$

ii $(7>=7)(8<5)$

iii $(6=3+3)(10<1)$

iv $(9=8)$

v $((6=3+3)(10<1))$

$1$