Discrete Structures Test 1

Truth table for ∼(p∧q)∨(p∨q)

(See hw 2 question 13)

truth table for p∧(∼q∨r)

(See hw 2 question 14)

h=john is healthy w= john is wealthy s=john is wise write the statement in logical form d) John is neither wealthy nor wise but he is healthy

(∼w∧∼s)∼h

(-15)+(-46)= do this in binary

-61₁₀ and 11010010 +11110001 ----------- 11000011₂

10111010 in base 10

-70₂

10011001 in base 10

-103₁₀

55 in binary

00110111₂

01010100-00010111=

00111101₂

4ADF83 in binary

010010101101111110000011

110111011+1001011010=

10000010101₂

123+(-94)= in binary

29₁₀ and 01111011₂ +10100010₂ ------------ 00011101₂

89+(-55)= Do this in binary and find the answer in base 10

34₁₀ and 01011001₂ +11001001₂ -------------- 100100010₂

1011011111000101 in hexadecimal

B7C5₁₆

Rewrite in 2 ways using the contrapositive Doing homework regularly is a necessary condition for jim to pass the course

If Jim didnt pass the course then he did not do his homework If Jim does his homework then he can pass the course

Contrapositive: If n is divisible by 6 then n is divisible by 2 and n is divisible by 3

If n is not divisible by 3 or n is not divisible by 2 then n is not divisible by 6

Write 2 if then statements, one being the contrapositive of the other: Sam will be allowed on signe's boat only if he is an expert sailor

If sam is allowed on Signe's boat then he is an expert sailor Sam is not an expert sailor if he is not allowed on signe's boat

Rewrite in 2 ways using the contrapositive: A necessary condition for this computer program to be correct is that it not produce error messages during translation

If the computer program did not produce error messages during translation then the computer program is correct. If the computer produced error messages during translation then the computer program is not correct

negation: If the decimal expansion of r is terminating then r is rational

If the decimal expansion of r is terminating and r is irrational

Rewrite in if then form Having two 45 degree angles is a sufficient condition for this to be a right triangle

If this triangle has two 45 degree angles then it is a right triangle

Contrapositive: If today is New Years eve, then tomorrow is January

If tomorrow is not january, then today is not new years eve

Converse and inverse If the decimal expansion of r is terminating then r is rational

Inverse) If the decimal expansion of r is not terminating then r is irrational Converse) If r is rational then the decimal expansion of r is terminating

Converse and inverse If today is New Years eve, then tomorrow is January

Inverse) If today is not new years eve then tomorrow is not january Converse) If tomorrow is not january then today is new years eve

Converse and inverse If n is divisible by 6 then n is divisible by 2 and n is divisible by 3

Inverse)If n is not divisible by 6 then n is not divisible by 2 or n is not divisible by 3 Converse) If n is divisible by 3 and divisible by 2 then n is divisible by 6

Use Demorgans laws to write negation for the statement Sam is an orange belt and Kate is a red belt

Sam is not an orange belt or Kate is not a red belt

Tautology or contradiction? (∼p∨q)∨(p∧∼q)

Tautology

Use Demorgans laws to write negation for the statement The train is late or my watch is fast

The train is not late and my watch is not fast

Determine whether the statements are logically equivalent (p→(q→r)) and (p→q)→r

They are not logically equivalent because one is different

Use Demorgans laws to write negation for the statement This computer program has a logical error in the first ten lines or it is being run with an incomplete data set

This computer program does not have a logical error in the first ten lines and it is not being run with an incomplete data set

(p∧~q)∧(~p∧~q) =(∼q∨p)∧(∼q∨~p) a =~q∨(p∧~p) b =~q∨c c =~q d

a)commutative b)distributive laws c)negation d)identity

Tautology or contradiction? ((∼p∧q)∧(q∧r)∧∼q

contradiction

h=john is healthy w= john is wealthy s=john is wise write the statement in logical form a) John is healthy and wealthy but not wise

h∧w∧∼s

Contrapositive: If the decimal expansion of r is terminating then r is rational

if r is irrational then the decimal expansion of r is not terminating

negation: If n is divisible by 6 then n is divisible by 2 and n is divisible by 3

n is divisible by 6 and n is not divisible by 2 or n is not divisible by 3

negation: If today is New Years eve, then tomorrow is January

negation of p→q is p∧∼q today is new years eve but tomorrow is January

(p→r)↔(q→r) truth table

question 10 homework 3

(p→(q→r))↔((p∧q)→r Truth table

see question 11 on homework 3

truth table for comparison between ∼(p∧q) and ∼p∧∼q

they are not logically equivalent because the truth table does not yield the same results for both

truth table for comparison between p∧c and p∨c

they are not logically equivalent because they don't come out with the same values in the truth table

h=john is healthy w= john is wealthy s=john is wise write the statement in logical form e) John is wealthy but he is not both healthy and wise

w∧∼(h∧s)

~(p∨~q)∨(~p∧~q)=~p find logical equivalences and supply a reason for each

~(p∨~q)∨(~p∧~q)≡~p Demorgan's (~p∧q)∨(~p∧~q)≡~p distribitive ~p∧(q∨~q)≡~p negation ~p∧t≡~p

h=john is healthy w= john is wealthy s=john is wise write the statement in logical form c)John is neither healthy, wealthy, nor wise

∼h∧∼w∧∼s

h=john is healthy w= john is wealthy s=john is wise write the statement in logical form b)John is not wealthy but he is healthy and wise

∼w∧h∧s