The document discusses predicates and propositional functions, emphasizing their roles in defining properties of variables. It explains how truth values are determined through examples and introduces logic circuits related to propositional logic. Additionally, it touches on formal and informal proofs, including proof by contradiction.

1. Discrete Structures
Logic and Proofs
Lecture-3

2. Clearly

6. Predicate
A Predicate or Propositional function is a
description of the property (or properties) a variable
or subject may have.
A proposition may be created from a propositional
function by either assigning a value to the variable or
by quantification.
The independent variable of a propositional function
must have a universe of discourse, which is a set
from which the variable can take values.
What is different
between proposition &
proposition logic…

7. Predicate (Conti…)
Let P(x) denote the statement “x > 3.” What are the
truth values of P(4) and P(2)?
Solution: We obtain the statement P(4) by setting x =
4 in the statement “x > 3.” Hence,
P(4), which is the statement “4 > 3,” is true.
However,
P(2), which is the statement “2 > 3,” is false.

26. Predicate Example
Let A(x) denote the statement “Computer x is
under attack by an intruder.” Suppose that of the
computers on campus, only CS2 and MATH1 are
currently under attack by intruders. What are truth
values of A(CS1), A(CS2), and A(MATH1)???

36. Logic Circuits and Propositional Logic
Problem: Build a digital circuit that produces the
output
(p ∨ ￢ r) ∧ ( ￢ p ∨ (q ∨ ￢ r)),
when given input bits p, q, and r.

37. Solution

38. Formal and informal proofs

47. Given p is true and the implication p → q is true then q is true

61. Reading Assignment
Proof by Contradiction