## Chapter - 1, Sentential Logic

July 18, 2015

### Summary

Statements with Variables. For eg: “x is divisible by 9”, “y is a person” are statements. Here x, and y are variables. These statements are true or false based on the worth of variables. Sets, a repertoire of objects. Bound and Unbound variables. Eg: ( y ∈ {x,vert,x^3 \$y\$ is a totally free variable, whereas \$x\$ is a bound variable. Free variables in a statement are for objects for which statement is talking about. Bound variables are just dummy variables to assist expush the concept. Hence bound variables dont represent any kind of object of the statement. The truth collection of a statement P(x) is the collection of all values of x that make the statement P(x) true. The set of all feasible worths of variables is call cosmos of discourse. Or variables range over this universe. In general, \$y ∈ x ∈ A,vert,P(x)\$ indicates the same thing as \$y ∈ A ∧ P(y)\$.

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### Solutions

Soln1

(a) \$D(6,3) land D(9,3) land also D(15, 3)\$ wbelow \$D(x, y)\$ implies \$x\$ is divisible by \$y\$.

(b) \$D(x,2) land also D(x,3) land lnot D(x, 4)\$ where \$D(x, y)\$ suggests \$x\$ is divisible by \$y\$.

(c) \$(lnot P(x) land P(y)) lor (P(x) land lnot P(y)\$ where \$P(x) = x in mathbbN,vert, x ext is prime \$.

Soln2

(a) \$M(x) land also M(y) land also (T(x,y) lor T(y,x))\$ where \$M(x)\$ is “x is men”, \$T(x, y)\$ indicates “x is taller than y”.

(b) \$<(B(x) lor B(y)) land also (R(x) lor R(y)>\$ wbelow \$B(x) ext and also R(x)\$ implies “x has brvery own eyes” and also “x has brown hairs” respectively.

(c) \$<(B(x) land also R(x)) lor (B(y) land R(y)>\$ wbelow \$B(x) ext and also R(x)\$ implies “x has actually brvery own eyes” and “x has brvery own hairs” respectively.

Soln3

(a) \$ x,vert,x ext is a world \$

(b) \$ x,vert,x ext is a university \$

(c) \$ x,vert,x ext is a state in US \$

(d) \$ x,vert,x ext is a district in Canada \$

Soln4

(a) < x^2,vert, x > 0 ext and x in mathbbN >

(b) \$ 2^x,vert, x in mathbbN \$

(c) \$ x in mathbbN,vert, 10 le x le 19 \$

Soln5

(a) \$−3 ∈ x ∈ mathbbRvert,13 − 2x > 1 Rightarrowhead -3 in mathbbR land also 19 > 1\$. No complimentary variables in the statement. Statement is true.

(b) \$4 ∈ x ∈ mathbbR^+vert,13 − 2x > 1 Rightarrowhead 4 in mathbbR^+ land also 5 > 1\$. No free variables in the statement. Statement is false.

(c) \$5 otin x ∈ mathbbRvert,13 − 2x > c Rightarrowhead lnot 5 in mathbbR land 3 > c Rightarrow 5 otin mathbbR lor 3 le c\$. One totally free variable(c) in the statement. (Thanks Maxwell for the correction)

Soln6

(a) \$(w ∈ mathbbR) land (13 - 2w > c)\$. Tright here are two totally free variables \$w\$ and also \$c\$.

(b) \$(4 in mathbbR) land also (13 - 2 imes 4 in P) Rightarrowhead (4 in mathbbR) land (5 in P)\$. The statement has actually no complimentary variables. It is a true statement.

(c) \$(4 in P) land (13 - 2 imes 4 > 1) Rightarrowhead (4 in P) land (5 > 1)\$. The statement has actually no free variables. It is a false statement.

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Soln7

(a) Conrad Hilton Jr., Michael Wilding, Michael Todd, Eddie Fisher, Richard Burton, John Warner, Larry Fortensky.

(b) \$ lor, land also, lnot \$

(c) Daniel Vellemale

Soln8

(a) 1, 3

(b) \$phi\$

(c)

Update:

As stated in comments, I acquired this wrong first time. Here is the correct answer: