Exam 1: The Foundations: Logic and Proofs

P(x, y) means $“ x + 2 y = x y "$ where x and y are integers. Determine the truth value of the statement. - $\forall x \exists y P ( x , y )$

Match the statement in symbols with one of the English statements in this list: 1. Some freshmen are math majors. 2. Every math major is a freshman. 3. No math major is a freshman. - $\forall x ( \neg ( M ( x ) \wedge \neg F ( x ) ) )$

suppose that $Q ( x ) \text { is “ } x + 1 = 2 x " \text {, }$ where x is a real number. Find the truth value of the statement. - $Q ( 2 )$

In questions , determine whether the proposition is TRUE or FALSE. -If 1 < 0, then 3 = 4.

Show that the premises "Jean is a student in my class" and "No student in my class is from England" imply the conclusion "Jean is not from England".

I will go to the play or read a book, but not both.

suppose the variables x and y represent real numbers, and L(x,y):x$\text { If } x < y \text {, then } x \text { is not equal to } y \text {. }$

suppose the variables x and y represent real numbers, and L(x,y):x0 P(x):x is a prime number. Write the statement in good English without using any variables in your answer. - $\forall x \exists y \quad L ( x , y )$

Prove that $( q \wedge ( p \rightarrow \neg q ) ) \rightarrow \neg p$ is a tautology using propositional equivalence and the laws of logic.

Match the statement in symbols with one of the English statements in this list: 1. Some freshmen are math majors. 2. Every math major is a freshman. 3. No math major is a freshman. - $\forall x ( M ( x ) \rightarrow \neg F ( x ) )$

Consider the following theorem: If x is an odd integer, then x + 2 is odd. Give a proof by contradiction of this theorem.

suppose the variable x represents people, and $F ( x ) : x \text { is friendly } \quad T ( x ) : x \text { is tall } \quad A ( x ) : x \text { is angry. }$ Write the statement using these predicates and any needed quantifiers. -If a person is friendly, then that person is not angry.

Write the contrapositive, converse, and inverse of the following: If you try hard, then you will win.

Find a proposition with three variables p, q, and r that is true when p and r are true and q is false, and false otherwise.

In questions , determine whether the proposition is TRUE or FALSE. -If 1 + 1 = 2 or 1 + 1 = 3, then 2 + 2 = 3 and 2 + 2 = 4.

P(x, y) means $“ x + 2 y = x y "$ where x and y are integers. Determine the truth value of the statement. - $\exists y \forall x P ( x , y )$

Find a proposition using only $p , q , \neg$ , and the connective $\vee$ with the truth table at the right. p q ?

Construct a combinatorial circuit using inverters, OR gates, and AND gates, that produces the outputs in from input bits p, q and r. - $( \neg p \wedge \neg q ) \vee ( p \wedge \neg r )$

Determine whether $p \rightarrow ( q \rightarrow r ) \text { and } p \rightarrow ( q \wedge r ) \text { are equivalent. }$

Explain why an argument of the following form is not valid: p\rightarrowq \negp \therefore\negq